quantum non-demolition measurements and quantum simulation · 2015-12-01 · all divincenzo...

Quantum non-demolition measurementsand quantum simulation

Thesis submitted to theFaculty of Mathematics, Computer Science and Physics

of the Leopold-Franzens University of Innsbruckin partial fulfillment

of the requirements for the degree of

Doctor rerum naturalium

by

Gerhard Kirchmair

Innsbruck, July 2010

Abstract

Quantum information processing (QIP) has developed into one of the “hot topics” in physics duringthe last decade. More and more experiments appeared aiming at the realization of a quantumcomputer and/or quantum simulations. Strings of ions stored in a Paul trap are currently one ofthe most promising systems for realizing quantum algorithms. The great challenges ion traps arefacing is the realization of robust high fidelity operations, scalable traps and the combination ofall DiVincenzo criteria in one system.

This thesis reports on the realization of simple QIP sequences and a quantum simulation withtrapped 40Ca+and 43Ca+ions. For both isotopes the quantum bit (qubit) is encoded in an opticaltransition. It consists of the ground state S1/2 and the metastable excited state D5/2. The isotope43Ca+has a nuclear spin of I = 7/2 and thus exhibits a very complicated level structure. Theadvantage of this isotope is that hyperfine levels can be used as a robust memory for quantuminformation.

In the framework of this thesis a high fidelity Mølmer-Sørensen gate operation was implemented.Two ion Bell states and three ion GHZ states were created with a respective fidelity of 99.3% and98.4% using 40Ca+ions. It was demonstrated that the Mølmer-Sørensen gate operation worksnearly as well for ions cooled to the ground state as for ions in thermal states of motion. Thefidelity achieved for thermal ions was still 98.4%. A thorough experimental analysis of the gatemechanism indicated possible error sources.

A combination of four DiVincenzo criteria (initialization, long coherence times, a universal set ofgates, readout) was realized by utilizing 43Ca+hyperfine states. Bell states, created on the opticalqubit, were stored in the qubit S1/2, F = 4, mf = 0 ↔ S1/2, F = 3,mf = 0 for 20 ms.

The experience gained with the entangling operation rendered it possible to do non-demolitiontwo-qubit measurements. The non-demolition measurements were used to experimentally testhidden variable theories, more precisely the Kochen Specker theorem. The experiments conductedshowed that hidden variables assuming non-contextuality of measurements cannot reproduce thebehavior of quantum systems. This was proven by violating an inequality akin to Bell tests. Theexperimental results also demonstrated the state-independence of the Kochen Specker argument.A possible compatibility loophole was closed by taking into account experimental errors in thetheory. The resulting modified equation was found to be violated in our experiments.

Finally, a proof of principle quantum simulation of the Dirac equation was implemented. Positionand momentum of the Dirac particle were mapped onto the respective quadrature components ofthe ion trap harmonic oscillator. A new method to measure the expectation value of position andmomentum was implemented. The particle trajectories obtained showed effects like Zitterbewegung.Using the high degree of control available in the experiment specific initial states were createdshowing different trajectories.

i

Abstract

ii

Zusammenfassung

Quanteninformationsverarbeitung (QIV) hat sich in eines der meist diskutierten Themen in derPhysik im letzten Jahrzehnt entwickelt. Immer mehr Experimente werden gebaut, die darauf abzie-len, einen Quantencomputer oder Quantensimulator zu realisieren. Besonders Ionen, gespeichertin einer Paul-Falle, gehoren zu den vielversprechendsten Systemen, um Quanten-Algorithmen zurealisieren. Die großen Herausforderungen, die es bei Ionenfallen zu losen gilt, sind die Realisierungvon Quantengattern mit hoher Gute, der Bau von skalierbaren Fallen und die Kombination allerDiVincenzo Kriterien in einem System.

Die vorliegende Arbeit beschreibt die Realisierung von einfachen Sequenzen zur QIV und zuQuanten-simulation mit 40Ca+ und 43Ca+ Ionen. Bei beiden Isotopen erfolgt die Kodierung desQuanten-Bits (Qubit) in einem sogenannten optischen Qubit, bestehend aus dem GrundzustandS1/2 und dem metastabilen angeregten Zustand D5/2 . Das Isotop 43Ca+ hat einen Kernspin vonI=7/2 und daher ein sehr kompliziertes Termschema. Der Vorteil dieses Isotops ist, dass HyperfeinNiveaus, sogenannte ”Uhrenubergange”, als robuste Speicher fur Quanteninformation verwendetwerden konnen.

Im Rahmen dieser Doktorarbeit wurde ein Mølmer-Sørensen Gatter hoher Gute realisiert. Eswurden mit diesem Gatter Bell-Zustande mit eine Gute von 99.3% und GHZ-Zustande, bestehendaus drei Ionen, mit einer Gute von 98.4% hergestellt. Weiters wurde gezeigt, dass das Mølmer-Sørensen Gatter auch mit Ionen funktioniert, die nicht in den Grundzustand gekuhlt wurden. Dieerreichte Gute unter diesen Voraussetzungen betrug immer noch 98,4%. Mogliche Fehlerquellender Gatter Operation wurden durch eine grundliche Untersuchung quantifiziert.

Durch Verwendung von 43Ca+ Hyperfein-Zustanden war es moglich, vier der funf DiVincenzoKriterien (Initialisierung, lange Koharenzzeiten, universeller Satz an Gattern, Zustands- Detektion)in einem System zu vereinen. Bell Zustande, die auf dem optischen Qubit erzeugt wurden, konntenins Hyperfein-Qubit S1/2, F = 4,mF = 0 ↔ S1/2, F = 3,mF = 0 ubertragen werden und fur mehrals 20 ms gespeichert werden.

Die mit den Gatter Operationen gewonnene Erfahrung ermoglichte die Realisierung von nicht de-struktiven Quantenmessungen. Diese nicht-destruktiven Messungen wurden dazu verwendet um so-genannte Theorien versteckter Variablen zu testen. Die durchgefuhrten Experimente zeigten, dassTheorien versteckter Variablen, aufbauend auf nicht kontextuellen Messungen, quantenmechanis-che Systeme nicht richtig beschreiben. Dies wurde gezeigt durch Verletzung einer Ungleichung sehrahnlich der Bellschen Ungleichung. Weiters ergaben die Messungen, dass die Verletzung der Un-gleichung unabhangig vom Quantenzustand ist. Das Kompatibilitatsschlupfloch wurde geschlossendurch Einbeziehen der experimentellen Fehler in die Theorie.

Letztendlich wurde eine Quantensimulation der Dirac Gleichung experimentell realisiert. Ortund Impuls des simulierten Teilchens wurden auf die betreffende Quadraturkomponente des har-

iii

Zusammenfassung

monischen Oszillators abgebildet. Eine neue Methode wurde verwendet, um den Erwartungswertdes Orts und Impuls des simulierten Teilchens zu bestimmen. Die gemessene Teilchentrajektorienzeigten relativistische Effekte wie Zitterbewegung. Durch Ausnutzen der außergewohnlichen exper-imentellen Kontrolle konnten verschiedene Anfangszustande erzeugt werden die unterschiedlicheTrajektorien zeigten.

Contents

Abstract i

Zusammenfassung iii

1. Introduction 1

2. Trapped calcium ions as qubits 5

2.1. Quantum computation and quantum bits . . . . . . . . . . . . . . . . . . . . . . . 5

2.2. Quantum harmonic oscillator and phase space . . . . . . . . . . . . . . . . . . . . . 10

2.3. Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Laser-ion interaction 19

3.1. Ion laser Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2. Non-resonant interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3. The S1/2 → D5/2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4. Microwave transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5. Mølmer-Sørensen gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6. Creating a coherent displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4. Experimental setup 33

4.1. Linear ion trap and electrode wiring . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2. Vacuum vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3. Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4. Optical access and single ion addressing . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5. Fluorescence detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6. Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.7. Computer control and RF generation . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5. Experimental techniques 47

5.1. Loading ions by photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2. Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3. Referencing the 729 nm laser to the ions . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4. Implementation of arbitrary qubit operations and ion shuttling . . . . . . . . . . . 52

5.5. State tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Contents

6. Entangling calcium ions 57

6.1. High fidelity two ion Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2. Three ion entangled states and arbitrary operations on three qubits . . . . . . . . 676.3. High fidelity entanglement of 43Ca+ hyperfine clock states . . . . . . . . . . . . . . 706.4. Comparison between 40Ca+ and 43Ca+. . . . . . . . . . . . . . . . . . . . . . . . . 74

7. Experimental test of quantum contextuality 77

7.1. The Kochen-Specker theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2. Experimental realization of QND measurements . . . . . . . . . . . . . . . . . . . . 817.3. Testing the Kochen Specker inequality . . . . . . . . . . . . . . . . . . . . . . . . . 837.4. Including imperfect measurements to close loopholes . . . . . . . . . . . . . . . . . 867.5. Experimental results on imperfect measurements . . . . . . . . . . . . . . . . . . . 89

8. Quantum simulation of the Dirac equation 91

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.2. The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928.3. Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

9. Summary and Outlook 101

A. The Kochen Specker measurements 105

A.1. Detailed results for the Kochen Specker measurements . . . . . . . . . . . . . . . . 105A.2. Hiding the second ion in the Kochen Specker measurements . . . . . . . . . . . . . 106

B. Methods for simulating the Dirac equation 107

B.1. Constructing a pure negative energy spinor . . . . . . . . . . . . . . . . . . . . . . 107B.2. Setting the phase of the Displacement pulse . . . . . . . . . . . . . . . . . . . . . . 107

C. Physical and optical properties of Calcium 109

D. Journal Publications 111

1. Introduction

Quantum mechanics (QM) was among the greatest developments in the twentieth century. Thefoundations of this theory were established around 1925 by Werner Heisenberg, Erwin Schrodinger,Max Born, Wolfgang Pauli, Niels Bohr, Albert Einstein, Paul Dirac and many others. Initiallyinvented for a better understanding of atomic spectra, quantum mechanics opened the door toa completely new field in physics. Nowadays it successfully explains chemical processes, materialbehavior and semiconductor devices, enabling the development of the modern computer technology.Computers in turn helped to numerically solve quantum mechanical problems which could not besolved analytically. In the following decades physicists realized that this approach has its limits asquantum systems consisting of only a few quantum bits (qubits) are already impossible to simulatewith current computer technology.

In the early 1980’s Paul Benioff [1] and Richard Feynman [2] came up with the idea of simulatingone quantum system by using another one. This so called quantum computer would only need anamount of qubits on the same order as the simulated system. Back in the 80’s these ideas whereconsidered as mere “Gedankenexperimente“ and not as realistic scenarios that could be developedinto experiments. It was considered impossible to use a single atom as a carrier for quantuminformation. Richard Feynman said in a lecture in 1986: ”. . . we are to be even more ridiculouslater and consider bits written on one atom instead of the present 1011 atoms. Such nonsense isvery entertaining to professors like me. I hope you will find it interesting and entertaining also.“

Today such procedures are experimental routine and quantum information processing as wellas quantum simulation are major research fields in theoretical and experimental physics. The biginterest was sparked by the discovery of a factorization algorithm on a quantum computer, whichis much faster than its classical counterpart, by Peter Shor in 1994 [3]. A large scale experimentalrealization of this algorithm would render public key encryption useless. Two years later LovK. Grover found that a quantum computer can search unsorted databases faster than a classicalcomputer [4]. Barenco et. al. [5] as well as DiVincenzo [6] showed in 1995 that an arbitraryquantum algorithm can be constructed with a limited set of single and two-qubit gates. This“universal“ set of operations is similar to the classical approach of constructing logic circuits outof NAND gates. To actually build a quantum computer certain prerequisites have to be met.These requirements were formulated by DiVincenzo [7] and are often called DiVincenzo criteria,according to which any quantum computer implementation requires

1. A scalable physical system with well characterized qubits

2. The ability to faithfully initialize the qubit to a simple well defined state

3. Long relevant coherence time, much longer than the gate operation time

4. A “universal“ set of quantum gates

1

Introduction

5. Qubit specific measurement capability

Later two additional requirements were added to incorporate quantum communication betweendistant nodes of a quantum network

6. The ability to interconvert stationary and flying qubits

7. The ability to faithfully transmit flying qubits between specified locations

Throughout the last decade research has shown thtat there are several experimental implementa-tions that fulfill most or even all of these points and might be considered as candidates for a quan-tum computer: nuclear spins [8], superconducting Josephson junctions [9, 10], quantum dots [11],neutral atoms [12], photons [13] and ions [14]. While nuclear spins are the most advanced systemwhen it comes to an actual implementation of algorithms and the number of qubits [15], they sufferfrom the inability to create pure states. This prevents scaling of this technology to higher numbersof quantum bits. Solid state implementations offer the promise of easy scalability akin to integratedcircuits as soon as one is able to reliably manufacture and control the basic building blocks. In thelast few years these devices made remarkable progress realizing Bell states [16], two-qubit gates [17]and quantum process tomography [18]. The limiting factor at the moment is the coherence time ofa few microseconds [19]. Recent experiments with neutral atoms demonstrated the extraordinarycontrol physicists have over single atoms [20] and ultracold ensembles [21]. These systems seem tobe good candidates for quantum simulations of solid state models [22].

One of the most promising systems for realizing a quantum computer are trapped ions [23]. Theyalready fulfilled three of the five DiVincenzo criteria even before they were considered as a systemfor quantum information processing (QIP): initialization [24], readout [25–27] and long coherencetimes [28]. Laser-cooled strings of ions had been observed[29–31] qualifying as well characterizedqubits which could be used as a quantum register. The trapping of charged particles in electricfields was first proposed by K.H. Kingdon in 1923 [32]. Radio frequency traps were developed byWolfgang Paul in the 1950’s and a single ion was trapped the first time by Neuhauser et al. in1980. The trapping of ions had a huge influence on atomic physics and sparked the developmentof optical clocks and mass spectrometers. Furthermore it opened up the possibility of controllingwell-isolated quantum systems for processing quantum information. This point was first noticedin the seminal paper by Peter Zoller and Ignacio Cirac in 1995 [14]. They proposed to use thecollective motion of an ion string as a bus system to couple individual ions. A series of laser pulsesacting on one ion at a time can be used to realize a two-qubit gate. This approach is also scalableas the resources increase only polynomially with the number of qubits. Within the same year atwo-qubit gate was demonstrated [33] with a single ion which set the starting point for quantuminformation processing with trapped ions.

In the last years ion trap experiments have shown a tremendous control of experimental param-eters. This control was the key to demonstrate the first steps towards a quantum computer and/orquantum simulations. Meanwhile all seven DiVincenzo criteria have been achieved in at leastproof of principle experiments. Some of the milestones are the investigation of several two-qubitgates [34–40], deterministic quantum teleportation with atoms [41, 42], quantum error correc-tion [43], entanglement purification [44], the creation of six [45] and eight ion entangled states [46]

2

and the implementation of simple algorithms like the Deutsch-Josza algorithm [47] and Groverssearch algorithm [48]. Along the way, several analysis techniques were developed like quantumstate tomography and quantum process tomography [49, 50]. The so called decoherence-free sub-space was investigated [51] and universal ion-trap quantum computation with decoherence-freequbits was realized [52]. Ions trapped in separate traps have been entangled [53] and quantumteleportation has been demonstrated [54] fulfilling the additional DiVincenzo criteria.

Today the big challenges on the road towards an ion trap quantum computer are the following:scaling current technology to a few ten or hundred qubits; combining all building blocks in onesystem; improving all operations to enter the fault tolerant regime. Scaling the trap technologyto a high number of ions is a difficult but important task as for a real commercially useful imple-mentation of Shor’s factoring algorithm some 105 qubits are needed. Several research groups arecurrently constructing and testing miniaturized traps but progress is slow. Recent experimentshave demonstrated the combination of the basic DiVincenzo criteria in one experiment [55, 56].

Entering the fault tolerant regime is important for realizing a large scale quantum computercapable of doing arbitrary long quantum algorithms. For this purpose quantum error correctionprotocols similar to classical computation are indispensable. The first protocols were discoveredby Peter Shor [57] and Andrew Steane [58]. Furthermore, Peter Shor showed that it is possible torealize quantum algorithms with arbitrarily small errors even when using imperfect operations [59].This is possible as soon as all operations enter the fault tolerant regime which means errors below10−2 . . . 10−4. The exact threshold depends on the overhead in the number of qubits one is willingto accept [60–62].

Even though all building blocks for an ion trap quantum computer have been demonstrated itwill still take time until a large scale functioning quantum computer will become reality. Never-theless, ion trap systems might soon outperform classical computers when it comes to quantumsimulations. Here the demands on the quality of the operations and the number of qubits is muchless. Decoherence might be even used to mimic the effect of an environment. Recently a proofof principle experiment showed the simulation of a quantum magnet [63], demonstrating a phasetransition from a paramagnetic to a ferromagnetic order with two-qubits. Another proof of prin-ciple experiment, the simulation of the free Dirac equation will be presented in chapter 8. So evenrelativistic quantum effects might be simulated with a quantum computer. Soon experiments willbe able to control and do simulations with 10-20 interacting qubits which is already a next toimpossible problem for a classical computer.

The high control over experimental parameters in ion traps opens up the possibility to dofundamental tests of QM. These test are important as it is still debated whether quantum mechanicscan be explained with so called hidden variable models. Hidden variables were introduced as somephysicists were unhappy with the Copenhagen interpretation of quantum mechanics which onlypredicts the probabilities for the outcome of a measurement. Another intriguing attribute of QMare its nonlocal features, e.g. a measurement of two entangled particles shows correlations which arebigger than classically possible. This was first discovered by Einstein, Podolsky and Rosen in theirfamous paper from 1935 [64]. This point lead physicists to the conclusion that QM is incompleteand other unknown physical phenomena beyond QM have to be included. In 1964 John Bell showedthat hidden variable theories have to be non local to explain all effects of quantum mechanics [65].

3

Introduction

He presented his famous Bell inequality which is satisfied by local hidden variables but violatedby QM. Several experiments have meanwhile proven that QM violates Bell’s inequality [66–69].Another class of hidden variable theories are non-contextual theories which are the subject of atheorem developed by Kochen, Specker and Bell [70–72] in the 60’s. Recent experiments [73–75],one of them presented in Chapter 7, have shown that also this model cannot explain QM. So farit seems there is no better theory than QM covering all observed phenomena.

The thesis is structured as follows: Chapter 2 contains a review of the main ideas and terms ofquantum information processing followed by a quick review of the quantum harmonic oscillator.The level structure of 40Ca+ and 43Ca+ ions is described and possible choices for qubits are illus-trated. Chapter 3 summarizes the interaction of the ions with electromagnetic fields with a specialfocus on the Mølmer-Sørensen entangling gate. Chapter 4 describes the apparatus consisting of thevacuum chamber, the lasers and the electronic control. Chapter 5 explains standard experimentaltechniques necessary for running the apparatus and realizing the experiments presented in the sub-sequent chapters. Chapter 6 describes the realization of an entangling operation working on 40Ca+

and 43Ca+ ions. Bell states with fidelities as high as 99.3% were created and the gate operationwas analyzed in detail. Furthermore we achieved an entanglement of 40Ca+ ions in thermal statesof motion and a mapping of the entanglement from the optical qubit onto the hyperfine qubitin 43Ca+Finally three-ion entangled states with 40Ca+ were created and the first block of a newscheme for universal quantum computation with trapped ions was implemented. Chapter 7 detailson a test of the non-contextual hidden variable theory proposed by Kochen, Specker and Bell in atwo ion system. Data are presented that demonstrate that QM measurements are contextual evenin the presence of experimental noise. Chapter 8 describes the realization of an experiment witha single ion simulating the free Dirac equation. The particle position measured as a function oftime shows Zitterbewegung and other relativistic effects. Chapter 9 concludes the thesis with asummary and outlook to future experiments.

The main findings of this thesis were published in the references [75–78].

4

2. Trapped calcium ions as qubits

This chapter will quickly review the mathematical notation for a two-level system and the basicideas of quantum computation. The mathematical excursus is continued by a description of a quan-tum harmonic oscillator with an emphasis on canonical variables and the phase space description.The energy level structure of the Calcium isotopes 40Ca+ 43Ca+ will be outlined and a number ofpossible implementations of a qubit in calcium ions will be shown.

2.1. Quantum computation and quantum bits

Inspired by [79] this section will present the mathematical description of a two-level system andwill give a brief summary on quantum computation. It will introduce the notions “qubit“ and“entanglement“ as well as single and two-qubit operations.

2.1.1. Qubits

For a real physical implementation of a quantum computer, quantum information has to be storedand manipulated. Akin to classical computation we will make use of two states of a system. Thesestates will represent the quantum states |↓〉 and |↑〉 instead of the logical 0 and 1. In trapped ions,a qubit can be realized by identifying two long-lived atomics states with the qubit states |↓〉 and|↑〉. The main difference from the classical case is that the system cannot only be in one of thetwo states but also in a linear superposition

|Ψ〉 = α |↓〉+ β |↑〉 (2.1)

with α and β complex numbers and the normalization condition |α|2 + |β|2 = 1. Such a two-levelsystem is called quantum bit/qubit in analogy to the classical case. The numbers |α|2 and |β|2represent the probability of measuring the qubit in state |↓〉 or |↑〉. A single measurement will justshow either |↓〉 or |↑〉. Only by many repetitions the full information about a quantum state canbe acquired.

A convenient and often used form of rewriting equation (2.1) is

|Ψ〉 = cos(

θ

2

)|↓〉+ sin

(θ

2

)eiφ |↑〉 (2.2)

where a global phase factor was dropped that is not observable in experiments. Thus a purequantum state is completely described by two real numbers θ and φ. This representation allows usto visualize the qubit’s state on sphere a with unit radius, where θ and φ are the rotation anglesof the Bloch vector. This sphere, called Bloch sphere, is shown in figure 2.1.

5

Trapped calcium ions as qubits

Figure 2.1.: Bloch sphere representation of a pure qubit state |Ψ〉 described by the rotationangles θ and φ.

The state description can be extended to N-qubits by using a set of basis states composed of the2N product states

|n〉 = |iN 〉 ⊗ . . .⊗ |i2〉 ⊗ |i1〉 (2.3)

with i ε 0, 1. With the vectors |n〉 we can now describe every possible state consisting of N qubitsas

ΨN =2N−1∑n=0

αn |n〉 (2.4)

where the state amplitudes αn have to satisfy the normalization condition∑2N−1

n=0 |αn|2 = 1.Unfortunately, the convenient Bloch sphere representation is not extendable to more than onequbit.

2.1.2. Quantum gates

A universal language to describe quantum algorithms is the quantum circuit model. In analogyto classical computing, where a computation is built up of logic gates, in this model a quantumcomputation consists of concatenations of elementary quantum operations acting on the qubits.It can be shown, that there exists a small set of single qubit operations and a universal two-qubitgate that can be used to perform arbitrary quantum algorithms [79].

The simplest interaction acting on an N-qubit system is a single qubit operation. These opera-tions have to preserve the norm of the state and can be conveniently described by using the Paulimatrices plus the Identity matrix

σx =

(0 11 0

), σy =

(0 −i

i 0

), σz =

(1 00 −1

), I =

(1 00 1

). (2.5)

In the Bloch sphere representation, single qubit operations can be described as a rotation aroundthe x, y, z axis or an arbitrary axis φ in the equatorial plane by an angle θ. The rotation matrixes

6

2.1 Quantum computation and quantum bits

are given by

U (j)x (θ) = e−i θ

2 σx =

(cos θ

2 −i sin θ2

−i sin θ2 cos θ

2

)(2.6)

U (j)y (θ) = e−i θ

2 σy =

(cos θ

2 − sin θ2

sin θ2 cos θ

2

)(2.7)

U (j)z (θ) = e−i θ

2 σz =

(e−i θ

2 00 ei θ

2

)(2.8)

U (j)(θ, φ) = e−i θ2 σφ =

(cos θ

2 −ie−iφ sin θ2

−ieiφ sin θ2 cos θ

2

), (2.9)

where σφ = cos(φ)σx + sin(φ) σy and j denote the ion subject to the rotation. Setting φ = 0in U (i)(θ, φ) results in a U

(i)x (θ) rotation and similarly setting φ = π/2 results in U

(i)y (θ). If we

can apply these rotations to every qubit of our quantum computer it is sufficient to introduce oneparticular type of multi-qubit operation to construct all arbitrary operations. There are manydifferent choices for universal gate operations. The best known example is the controlled -NOT(CNOT) operation where the state of the target qubit is flipped or not depending on the state ofthe control qubit. The matrix representation of this operation is

UCNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

(2.10)

where the matrix is notated with respect to the basis order |↓↓〉 , |↓↑〉 , |↑↓〉 , |↑↑〉. Another im-portant two-qubit operation is the Mølmer-Sørensen [80, 81] (MS) gate operation. This gate wasimplemented in this thesis and will be further discussed theoretically in chapter 3 and experimen-tally in chapter 6. Its matrix representation is

UMSx

(π

2

)= e−i π

4 σx⊗σx =1− i

2

1 0 0 i

0 1 −i 00 −i 1 0i 0 0 1

. (2.11)

This gate introduces collective spin flips on two ions and can be extended to an N-qubit gate(see chapter 3.5). The normalization factor in front of the matrix is required because two-qubitoperations have to preserve the norm such that U†

gateUgate = I. The Mølmer-Sørensen gate iscompletely equivalent to a CNOT operation up to some single qubit rotations.

2.1.3. Quantum state measurement

At the end of a quantum algorithm/simulation during which the system evolved according to aunitary operation the outcome needs to be determined. This requires a detection of the qubit statesby a projective measurement. Its experimental implementation for the case of trapped ions will be

7


discussed in chapter 4.5. Mathematically this process can be described by a set of measurementoperators Mm such that the results m of a measurement occur with a probability

p(m) = 〈Ψ|M†mMm |Ψ〉 . (2.12)

The state of the system after the measurement is one of the eigenstates of the measurement operator

∣∣∣Ψ⟩

=Mm |Ψ〉√

〈Ψ|M†mMm |Ψ〉

(2.13)

The natural measurement basis for trapped ions are the projectors onto the energy eigenstates.Thus one can only measure the σz components of every qubit state Ψ by a simple fluorescencedetection. If projections onto other spin components A = ~n~σ are of interest (where ~n is thedirection of the desired spin component) a modified technique has to be employed.

The measurement basis can be changed by applying a unitary state transformation U prior tothe state detection on each qubit individually. This means the subspace H+

A = ψ|Aψ = ψ ismapped onto the subspace H+

σz= ψ′|σzψ

′ = ψ′. Thus a σz measurement on the transformedstate |Ψ′〉 = U |Ψ〉 is equivalent to a measurement of the desired observable A on the original state|Ψ〉. The unitary state transformation U satisfies

A = U†σizU (2.14)

and is given by a rotation around an axis on the Bloch sphere that is both perpendicular to ~ez andto ~n. The final inverse operation U† maps the projected state onto an eigenstate of the observableA which is in general not necessary. Nevertheless it becomes important when one is interested indoing non-demolition measurements in succession (see chapter 7). The mapping on an eigenstateensures that subsequent commuting measurements will still determine the right value.

An additional complication appears if e.g. a spin correlation like σz ⊗ σz, the so called parity,of a two ion state should be measured. If exactly this observable should be determined in a non-demolition measurement it is necessary to apply entangling two-qubit gates together with singlequbit operations, that map this information on a single qubit. An example for such an operationU is a CNOT gate which will map the parity of the two ions on the state of the target ion.

initial state mapped state inital state mapped stateσz ⊗ σz I ⊗ σz

|↓↓〉 |↓↑〉 (-1) · (-1) = +1 +1|↓↑〉 |↓↓〉 (-1) · 1 = -1 -1|↑↓〉 |↑↓〉 1 · (-1) = -1 -1|↑↑〉 |↑↑〉 1 · 1 = +1 +1

By measuring only the state of the second ion after the mapping, it is ensured, that the paritymeasurement reveals a single bit of information. It is important to note, that such a measurement isnot equivalent to measuring σz on every ion and correlating the results afterwards to get σz ⊗ σz.In the latter case one would not only get the information whether the ions are in the same oropposite spin state but also in which state each ion is.

8

2.1 Quantum computation and quantum bits

2.1.4. Entanglement

Called Spooky Action at a Distance [64] by Einstein, Podolsky and Rosen, entanglement brings upsome of the most striking features of quantum mechanics. Let us consider an entangled state fortwo qubits, the so called Bell state

|Ψ〉 =|↑↑〉+ |↓↓〉√

2(2.15)

which was created in several experiments described in this thesis (see chapter 6). Whenever thestate of one qubit of this entangled state is measured, one will find a perfectly correlated resultwhen measuring the other. If the first qubit is in the state |↓〉 /(|↑〉) the second qubit will be in|↓〉 /(|↑〉) or vice versa so both qubits will always be found in the same state. This behavior willviolate one or both of the assumptions Einstein and Co. made for a physical realistic theory :

• Realism: The assumption that the physical properties of a system have definite values whichexist independent of observation

• Locality : The assumption that a measurement on one particle does not influence the resultof the measurement on the second particle.

These assumptions seem to be intuitively plausible but nevertheless experiments conducted withthese states [66] show that they disobey the so called Bell inequality [72]. This then leads tothe conclusion that they have to disobey one or both of the upper points. There is still a lot ofdiscussion going on about the results of these experiments that go beyond the scope of this work.More information about this subject can be found in references [82–84].

A more rigorous criterion for entangled states can be phrased as: A state is entangled if itscomposite wavefunction cannot be written as a product of wavefunctions of the subsystems. Math-ematically this means

|ΨAB〉 6= |ΨA〉 ⊗ |ΨB〉 . (2.16)

Creating entangled states is nowadays a standard procedure which can be employed in severaldifferent physical systems [46, 85–87]. A single application of a Mølmer-Sørensen-gate to |↓↓〉creates already an entangled state of the form shown in equation (2.15) They can be used asresources for secure quantum communication [88] or certain types of quantum computation [89].Entanglement seems to appear in most quantum algorithms however it is still a matter of debateif the power of an algorithm comes from entanglement or not.

In chapter 7 of this thesis another peculiar feature of quantum-mechanics, the so called contex-tualtity of measurements, will be discussed. Contextualtity of measurements means that certaintypes of measurements influence another even though they should not. The states measured inheritentanglement not from the beginning but entanglement will be created during the measurementprocess. Entanglement plays a central role in quantum mechanics.

2.1.5. Density matrix description

So far the description of a qubit is sufficient for pure states but fails for states which are mixedor not completely known. Here the density operator language provides a convenient means todescribe these systems. All postulates for quantum mechanics regarding state description, state

9


evolution and measurement can be reformulated in terms of density operators [79]. For a systemwhich can be in pure states Ψi with a probability pi the density operator is given by

ρ =∑

i

pi |Ψi〉〈Ψi| . (2.17)

One has to note that the probabilities pi are only well defined for an ensemble of states. A densitymatrix has to satisfy the conditions

1. the trace of ρ is equal to one

2. ρ is a positive operator.

For a d-dimensional quantum system it is possible to expand the density matrix ρ using a basisof d2 hermitian matrices that are mutually orthogonal. A handy choice for a single qubit are thePauli matrices together with the identity

ρ =12

∑

i=x,y,z,I

λiσi. (2.18)

The parameters λi which completely characterize the state are given by

λi = Trρσi. (2.19)

So each of these parameters corresponds to the outcome of a projective measurement determiningthe expectation values of the Pauli operators 〈σi〉. This will become handy for reconstructing thequantum state in the experiment with a technique called state tomography (chapter 5). As one cansee from equation (2.20) it is sufficient to determine all λi to get the full knowledge of a quantumstate. The parameters λi uniquely define a vector on the Bloch sphere (similar to figure 2.1) butnot only the angles but also the length of the vector is now of importance. The length of the vectoris given by

∑x,y,z λ2

i and it shows the purity of a state. A pure state has a unit length vector, apartially mixed state has a vector with a length smaller than 1 and a completely mixed state hasa vector with a length equal to zero.

This description can be straightforwardly extended to multi-qubit systems. The density matrixfor a N-qubit system is written in terms of tensor products of the Pauli matrices

ρN =1

2N

∑

i1,i2...iN ,=x,y,z,I

λi1,i2...iNσi1 ⊗ σi2 ⊗ . . .⊗ σin . (2.20)

Normalization requires that λI,I...I = 1 such that 4N−1 parameters define the state in the Hilbert-space. Each mi1,i2...iN

corresponds again to a projective measurement determining 〈σi1 ⊗ . . .⊗ σin〉.

2.2. Quantum harmonic oscillator and phase space

An ion in a trap is one of the best model systems for a perfect quantum harmonic oscillator.Usually the harmonic oscillator is only used to mediate interactions between the ions but is not

10

2.2 Quantum harmonic oscillator and phase space

regarded by itself as an interesting system. In chapter 8 it will be shown that using a single ionin the harmonic oscillator of the trap one can conduct a simulation of the Dirac Equation. Theinformation of the position and the momentum of the Dirac particle will be encoded in the positionand momentum observables of the harmonic oscillator. Hence this section will reviwe the analysisof a quantum harmonic oscillator with an emphasis on the phase space description. At the end aclassical driving force will be introduced and the effects on the state of the oscillator will be shown.This chapter is strongly inspired by [37, 90, 91].

2.2.1. Harmonic oscillator hamiltonian

A quantum harmonic oscillator is a particle of mass m moving in a 1-d quadratic potential V (xc) =mω2x2

c/2. Classically the state of a harmonic oscillator is defined by its position and momentumcoordinates xc and pc. A classical harmonic oscillator is best described in phase space which isgiven by the complex plane spanned by the variable xc + ipc. The motion of the particle can bedescribed by a point in phase space which is rotating with an angular frequency ω along an ellipsecentered at the origin. In quantum mechanics the position and momentum will be replaced by theobservables x and p. They obey the commutation relation

[x, p] = i~. (2.21)

With these operators the Hamiltonian for a quantum harmonic oscillator reads as

Hm =p2

2m+

12mω2x2. (2.22)

Introducing the non-hermitian operator a and its conjugate a† such that

x = (a + a†)∆ and p = ia† − a

2~∆

(2.23)

with the characteristic length ∆ =√~/(2mω) one can rewrite equation (2.22) and arrive at the

well known quantized version of the harmonic oscillator Hamiltonian

Hm = ~ω(

a†a +12

). (2.24)

The spectrum of this Hamiltonian is a ladder of equidistant energy levels EN with a spacing of~ω starting at the ground state energy of E0 = ~ω/2. The corresponding wavefunctions |N〉 arecalled Fock States and will obey the following relations

a† |n〉 =√

n + 1 |n + 1〉 (2.25)

a |n〉 =√

n |n− 1〉 (2.26)

11


Re a

Im aD

h2D| |a

Re a

Im a

wt

a b

Figure 2.2.: (a)Phase space representation of a coherent state |α〉. The coordinates of thecenter of the disk correspond to 〈x〉 and 〈p〉. The size of the disk represents the variance ofthe state here with unit length along p and x. (b) Time evolution of a coherent state. Thestate rotates with the trap frequency ω around the origin.

and

a†n |0〉 =√

n! |n〉 (2.27)

a†a |n〉 = n |n〉 . (2.28)

Often a and a† are called lowering and raising operators, because they can be used to climb upand down the ladder of states. The spatial extension of the ground state |0〉 is given exactly by ∆.

2.2.2. Driven quantum harmonic oscillator

If a harmonic oscillator is driven by a classical force

F (t) = A sin(ωdrivet + φ) =A

2i(ei(ωdrivet+φ) − e−i(ωdrivet+φ)) (2.29)

the Hamiltonian is given byHforce = −xF (t). (2.30)

The action of this force on the ground state of the harmonic oscillator can be best explained bychanging into the interaction picture. A suitable transformation to do this is given by U |Ψ〉 = |Ψ′〉with U = eiHmt/~. The Hamiltonian in the interaction picture then reads as

Hiforce = −∆(ae−iωt + a†eiωt)F (t). (2.31)

For ω = ωdrive a resonance appears and using the rotating wave approximation the Hamiltonianreduces to

Hiforce = −A∆

2i(ae−iφ + a†eiφ) (2.32)

The evolution of the ground state under the action of this Hamiltonian is given by the unitaryoperator

D(α) = eα∗a−αa† (2.33)

12

2.3 Atomic Structure

with α = −A∆t/(2~)e−iφ. This propagator is a so called displacement operator. Applying it tothe ground state of motion creates the coherent state |α〉

D(α) |0〉 = |α〉 = e−|α2|/2

∑n

αn

√n!|n〉 . (2.34)

This can be generalized to cover the displacement of a coherent state |β〉 by using the Baker–Campbell–Hausdorff relation D(α)D(β) = D(α + β)ei Imαβ∗

D(α) |β〉 = D(α)D(β) |0〉 = ei Imαβ∗|α + β > . (2.35)

A coherent state obeys the following relations

a |α〉 = α |α〉 (2.36)

〈α| a† = α∗ 〈α| (2.37)

and

〈α| x |α〉 = 2∆ Reα (2.38)

〈α| p |α〉 =~∆

Imα. (2.39)

The real part of α corresponds to the mean position while the imaginary part corresponds tothe mean momentum. A convenient way to display these states is a phase space diagram (seefigure 2.2(a)) with α = 〈x〉 + i 〈p〉 with x in units of 2∆ and p in units of ~/∆. The variance forposition and momentum is also given by ∆ as

δx = ∆ and δp =~

2∆. (2.40)

A coherent state is the closest analogy to classical states one can achieve for a quantum harmonicoscillator and is hence called quasi-classical state. The state evolution, a rotation in phase space (seefigure 2.2(a)), cannot be seen in ion trap experiments because the laser creating the displacementacts as a reference oscillator rotating with the coherent state. How to create a displacementoperator with laser fields will be explained in chapter 3.

2.3. Atomic Structure

There are several ways to experimentally realize a good approximation of a two-level system.Ideally the ion has to be completely separated from the environment and should only interact withthe laser. Experimentally this means, that the laser-ion coupling has to be much larger than anycoupling to the environment. A suitable choice for a qubit are either hyperfine/Zeeman groundstates or states connected by a forbidden transition. The element used to realize a qubit in thisthesis is the single charged alkali earth ion Ca+. This section describes the relevant level structureof the two isotopes 40Ca+ and 43Ca+ as well as the qubits realized in these ions. A more detaileddescription of 40Ca+ can be found in [92] and [93] whereas for 43Ca+ the reference is [94].

13


Figure 2.3.: Detailed energy level scheme showing all Zeeman sublevels of the three lowestorbitals of the 40Ca+ ion. Laser light at 397 nm is used for Doppler-cooling, optical pumpingand detection. The lasers at 866 nm, 850 nm and 854 nm pump out the D-states. Anultra-stable laser at 729 nm is used for manipulating the qubit encoded in the quadrupoletransition, state preparation, optical shelving and ground-state cooling. The qubit is encodedin the states S1/2(mj = 1/2) and D5/2(mj = 3/2). The wavelength in air, natural lifetimesτ and the branching fractions (denoted as XX% for the respective transition) are taken fromreferences [95–98].

Level scheme, Zeeman splitting and qubit choice

The coarse level structure for 40Ca+ and 43Ca+ is the same. The energy level schemes showingthe lowest three orbitals of the valence electron for 40Ca+ and 43Ca+ can be seen in figure 2.3and figure 2.4. The S1/2 ground state is connected via a 397 nm dipole transition with the P1/2

state and a 393 nm dipole transition with the P3/2 state. The P-levels both having a lifetime ofabout 7 ns exhibit a branching ratio such that one out of 14 decays populates the D-levels.1 Toavoid population trapping in the long lived D-states one can use the dipole transitions at 854 nm,850 nm and 866 nm to pump out these levels. These wavelengths connect the two D-states toeither a P1/2 or P3/2 level via a dipole transition. A quadrupole transition at 729 nm connects theS1/2 and the D5/2 level. Since this transition is dipole-forbidden, the D5/2 level is metastable andhas a lifetime of about 1.2 s which sets an upper time limit for the usefulness of a qubit encodedin this state.

Calcium 40

In a magnetic field all levels split up into sublevels due to the Zeeman effect (see figure 2.3). Thesplitting of the states is a few MHz for the magnetic fields typically used in experiments. Thus itis well resolved for the quadrupole transition. For all the other transitions the splitting is hiddenwithin the linewidth of the transition. The S1/2 ground state splits up in two Zeeman sublevels withmS = ±1/2 and the D5/2 level splits up in six Zeeman sublevels with mD = ±1/2,±3/2,±5/2. For

1An exact measurement of the P3/2 branching ratio measurements using QIP methods can be found in reference [95].

14


P1/2

P3/2

D5/2

D3/2

S1/2

729 nm

397 nm

866 nm

854 nm

2 317.3(1.7)3 178.0(0.8)4 -10.1(0.9)5 -249.3(1.2)

3 1814.4046611(18)

4 -1411.2036253(14)

F (MHz)dhfs

3456

1 41.5267(6)2 34.9516(4)

24.6348(2)10.0312(3)-9.5858(3)-35.1246(4)

F (MHz)dhfs

345

2 205.6(1.6)117.5(0.8)

-164.5(1.1)-4.5(0.8)

3

4

327.2(2)

-254.5(2)

Figure 2.4.: 43Ca+ level scheme showing the hyperfine splitting of the lowest energylevels. Hyperfine shifts δhfs of the levels are quoted in MHz (the splittings are taken from[94, 99, 100]). Laser light at 397nm is used for Doppler-cooling, optical pumping and statedetection. The lasers at 866 nm, 850 nm and 854 nm pump out the D-states. An ultra stablelaser at 729 nm is used for manipulating the qubit encoded in the quadrupole transition,state preparation, optical shelving and ground-state cooling.

a quadrupole transition changes in the magnetic quantum number of ∆m = 0,±1,±2 are allowedby the selection rules. This results in a total of ten possible transitions from S1/2 to D5/2.

One of these pairs of levels, the S1/2,mj = 1/2 together with the D5/2,mj = 3/2 was used asa qubit for the experiments described in this thesis. The only exception to this choice are someof the experiments presented in chapter 6.1 where the S1/2, mj = 1/2 level together with theD5/2,mj = 5/2 level was used. These kinds of qubit are often termed optical qubits because thetransition frequency lies in the optical domain. As already mentioned, the sensitivity of the qubiton the environment, here especially magnetic field fluctuations, is of importance. The dependenceof the transition frequency on the magnetic field B is given by

∆ωS→D = µB(ms gS1/2 −mD gD5/2)B (2.41)

where µB denotes the Bohr magneton and gS1/2/gD5/2 the Lande-g-factor of the S1/2/D5/2 level.In the best case the lowest sensitivity one could get with respect to changes in the magneticfield is 560kHz/G for the ∆m = 0 transitions which are usually used as a qubit. The employedS1/2,mj = 1/2 → D5/2,mj = 3/2 transition has a factor of two higher dependence but comes withthe advantage that the coupling strength can be maximized by properly setting the polarizationand direction of the laser light with respect to the magnetic field (see chapter 3). This effectivelyreduces the time needed for operations.

15


Calcium 43

The isotope 43Ca+ is the only stable calcium isotope with a non-zero nuclear spin. It has a nuclearspin of I = 7/2. Its level scheme is shown in figure 2.4. The coupling between electronic angularmomentum J and nuclear spin results in a hyperfine structure. The total angular momentum Fcan take on the values |J − I| ≤ F ≤ |J + I|. For the ground state of 43Ca+ we get two hyperfinestates with F=3 and F=4 split by a frequency difference of 3.225 GHz [99]. The D state withJ = 5/2 consists of 6 hyperfine states ranging from F=1 to F=6. In a magnetic field all hyperfinelevels split up in 2F + 1 Zeeman states.

This huge state manifold opens up several possibilities to encode a qubit. Two choices areimplemented in the experiments conducted in this thesis. An encoding in the two hyperfine groundstates termed hyperfine qubit or, similar to 40Ca+, an encoding using an optical qubit.

The first choice is interesting because there is no limitation by spontaneous decay. Furthermorewe can get a qubit which is insensitive to magnetic field fluctuations by choosing the mF = 0sublevels of the two hyperfine ground states. This particular qubit was implemented in this thesis(see chapter 6). From the Breit-Rabi formula [101] one can directly obtain that these sublevelsexhibit no first order Zeeman shift. However the linear dependence vanishes only for zero magneticfield. For a finite field the differential shift is given by a second order approximation as [94]

∆ω0→0 =(gJ − gI)2µ2

B

2~∆EhfsB2 ' 2π · 1.2 kHz/G2 ·B2 (2.42)

with the nuclear g-factor gI accounting for the complex structure of the nucleus and the Landefactor

gJ ' 1 +J(J + 1) + S(S + 1)− L(L + 1)

2J(J + 1). (2.43)

This means that for a typical magnetic field of 4 G the magnetic field sensitivity is 19.2 kHz/Gwhich is much lower than for the best transition in 40Ca+. An additional advantage of hyperfinequbits is that the transition frequency is in the GHz range so that the transition can be directlyaddressed with a µ-wave field. Stable frequency sources with computer controlled power, phaseand frequency are readily available. If additional single ion addressing is required, one can switchto a Raman laser setup as the microwave field cannot be focused. The two laser beams for theRaman setup can be derived from the same laser source. The frequency splitting is achieved byfrequency shifting the light fields with acousto-optical modulators which ensures phase coherence.An example for the implementation of a Raman laser with 43Ca+ ions can be found in [94] and [102].In this thesis the hyperfine qubit is solely manipulated by a microwave field.

The second choice, an optical qubit manipulated on the quadrupole transition is also appealing.Although one is still limited by the finite lifetime of the D-state one can find transitions whichare insensitive to magnetic field fluctuations. This is due to the additional coupling of the nuclearmagnetic moment to the electron spin and the magnetic field which modifies equation (2.41). Thetransition frequencies depend in a nonlinear way on the magnetic field which gives rise to a numberof different transitions which are insensitive to magnetic field fluctuations at a non zero magneticfield. A good example is the S1/2, F = 4,mF = 4 → D5/2, F = 4, mF = 3 transition whichbecomes field insensitive at 3.38 G and 4.96 G with a second-order Zeeman shift of ∓16kHz/G2.

16


A much more detailed explanation of the hyperfine splitting in 43Ca+ and the interaction with staticmagnetic fields is given in [94]. Although the optical qubit in 43Ca+ is insensitive to magnetic fieldvariations one should not forget that the coherence on all optical qubits is still limited by theline width of the laser mediating the coupling. With the currently available quadrupole laser thecoherence time for a single ion is limited to about 8 ms (see chapter 4.6.2).

In the experiment two different qubits on the quadrupole transition were realized. One qubitwas realized using the levels S1/2, F = 4,mF = 0 ↔ D5/2(F = 6,mF = 1) at a magnetic field of6 G. The second qubit was realized using the levels S1/2, F = 4,mF = 0 ↔ D5/2(F = 4,mF = 2)at a magnetic field of 3.4 G. Both were chosen because they are separated from other transitions byseveral MHz, have a moderate magnetic field dependence of 350 kHz/G and 220 kHz/G and highcoupling strength. Furthermore, their coupling strength can be maximized by properly setting thepolarization and direction of the laser light with respect to the magnetic field (see chapter 3).

17


18

3. Laser-ion interaction

This chapter summarizes the interactions between an ion trapped in a Paul trap and an incidentlaser beam. The first two parts review resonant and non-resonant single-qubit interactions. Thefollowing sections describe the interactions relevant for a quadrupole transition and a microwavetransition. Then the focus shifts to the implementation of a two-qubit operation by making useof a bichromatic light field. This part was also published in [76]. Finally the bichromatic lightfield will be used to create coherent displacement forces acting on the ion trap harmonic oscillator.Furthermore, the bichromatic light field is used to measure the expectation values of position andmomentum for the harmonic oscillator.

3.1. Ion laser Hamiltonian

The Hamiltonian describing the trapped ion consists of two parts. The first part describes the ionthat can be treated as a two-level system with a laser tuned close to this transition. The second partdescribes the ion in the external confining potential of the trap that forms a harmonic oscillator.In the following description we assign a pseudo-spin to the qubit. The qubit states can then beidentified with |↓〉 (|↑〉) for the lower(upper) energy level. For an ion with mass m oscillating withfrequency ω in the harmonic potential the system can be described by the following Hamiltonians

Hm =p2

2m+

12mω2x2 (3.1)

He =12~νσz (3.2)

Hl = ~Ω(σ+ + σ−)(ei(kx−νlt+φ) + e−i(kx−νlt+φ)) (3.3)

where σz, σ+ = 1

2 (σx +iσy), σ− = 12 (σx−iσy) are Pauli spin operators and ν denotes the frequency

splitting of the qubit levels. Furthermore, Ω includes all details about the exact form of the atomlight interaction and is called the Rabi frequency, while k is the wave vector, νl the laser frequencyand φ the phase of the light field. Hm describes the motion of the ion in the harmonic potential,while He describes the ions internal state. The laser-ion interaction is contained in the HamiltonianHl. Using the creation and annihilation operators a†, a the Hamiltonians Hm and Hl can bereexpressed as

H0 = Hm + He = ~ω(a†a +12) +

12~νσz (3.4)

Hl = ~Ω(σ+ + σ−)(eiη(a+a†)e−i(νlt−φ) + e−iη(a+a†)ei(νlt−φ)) (3.5)

19

Laser-ion interaction

Figure 3.1.: Combined energy levels of a two-level system and a harmonic oscillator. Theblack arrows indicate carrier transitions whereas the red/blue arrows indicate the respectivesideband transitions.

Here the Lamb Dicke parameter η is used, which is defined as

η = ~k~ez

√~

2mω(3.6)

where ~ez denotes the oscillation axis. The Lamb Dicke parameter relates the spatial extension ofthe harmonic oscillator ground state to the wavelength of the atomic transition. It describes theability of the light field to couple to the motion of the ion in the harmonic oscillator.

Transforming H = H0 + Hl into the interaction picture with Hint = U†HU , U = eiH0t/~ andusing the rotating wave approximation leads to the Hamiltonian

Hint(t) = ~Ω(σ+eiη(ae−iωt+a†eiωt)e−i((νl−ν)t−φ) + h.c.). (3.7)

Depending on the detuning ∆ = νl − ν of the laser from the transition frequency ν, it will couplecertain electronic and motional states. If the detuning is exactly ∆ = s ω it will couple the state|↓, n〉 to |↑, n + s〉, where n is the vibrational quantum number and s ∈ N0. Equation (3.7) containsall details about the laser ion interaction.

3.1.1. Lamb Dicke regime

An ion is said to be in the Lamb Dicke regime if the extension of its wavefunction is small comparedto the wavelength of the light. In this regime, the inequality η2xrms ¿ 1 is fulfilled with xrms theextent of the vibrational mode’s wave function. Then equation (3.7) can be simplified by a Taylorexpansion of

eiη(ae−iωt+a†eiωt) = 1 + iη(ae−iωt + a†eiωt) + O(η2). (3.8)

Only first order terms need to be considered, that is transitions which change the vibrationalquantum number by no more than ±1. The interaction Hamiltonian is now given by

Hint(t) = ~Ωσ+(1 + iη(ae−iωt + a†eiωt))e−i((νl−ν)t−φ) + h.c.. (3.9)

This Hamiltonian gives rise to three different resonances:

• For νl = ν only the electronic state of the ion is altered while ∆n = 0. The transition(|↓, n〉 → |↑, n〉 is called carrier transition where the coupling strength, including a second

20

3.2 Non-resonant interactions

order term, scales as Ωn,n = (1− η2n)Ω. The small correction ≈ n is obtained going beyondthe approximation (3.9). The effective Hamiltonian is given by

Hcar(t) = ~Ωn,n(σ+eiφ + σ−e−iφ). (3.10)

Depending on the angle φ one can implement σx, σy interactions realizing two of the threesingle qubit operations described in chapter 2.1.2.

• For νl = ν − ωz the electronic state of the ion is changed and n decreases by one. The lasercouples the states |↓, n〉 → |↑, n− 1〉. The coupling strength for this red sideband transitionscales as Ωn−1,n = η

√nΩ. The Hamiltonian is given by

Hred(t) = i~Ωn−1,n(aσ+eiφ − a†σ−e−iφ). (3.11)

• Similar for νl = ν + ωz the electronic state of the ion is changed and n increases by one.The laser couples |↓, n〉 → |↑, n + 1〉. The coupling strength for this blue sideband transitionscales as Ωn+1,n = η

√n + 1Ω. The Hamiltonian is given by

Hblue(t) = i~Ωn+1,n(a†σ+eiφ − aσ−e−iφ). (3.12)

A graphical illustration of carrier and sideband transitions is shown in figure 3.1. It shows thecombined energy levels of a two-level system and a harmonic oscillator.

With the help of sideband transitions one can convert the internal excitation of one ion into acollective motion of the whole ion string. For example the state |↑↓ . . . ↓, 0〉, carrying one electronicexcitation, will be transformed into the state |↓↓ . . . ↓, 1〉, carrying one phonon, by a π-pulse onthe red sideband. This excitation can be transferred onto any other ion in the string by anotherred sideband pulse. In this way two or more ions can interact with another. The motion acts as akind of bus system mediating the interaction.

3.2. Non-resonant interactions

So far we have only considered resonant interactions. However, also non-resonant interactionsplay a crucial role in the ion-light interaction. On one hand, they can be used to create desiredinteractions, on the other hand they cause unwanted effects. Non-resonant interactions becomeespecially important when driving sideband transitions. Due to the small coupling of the sidebandsto the laser a lot of light is needed. This causes a frequency shift on the atomic levels due to thecarrier and population transfer on the carrier.

3.2.1. Light shifts

A light field which is off-resonant from a transition will shift the atomic levels. This effect is knownas AC-Stark-effect. Consider a two-level atom with an unperturbed energy difference E = ~ν anda detuning of the light field by δ = ν − νl from this transition. The amplitude of the light fieldis given by a Rabi frequency Ω. The level shift is calculated by going along the same line of

21


Figure 3.2.: (a) Rabi oscillations on the sideband (red) with a Rabi frequency Ω =2π ·45 kHz. The black line with an amplitude of 0.025 are Rabi oscillations with a frequencyof 1.2 MHz taking place on the carrier transition. The oscillation frequency and amplitudeis given by equation (3.15). (b) Pulse form of a shaped pulse. The slopes are shaped as aBlackman window, the figure shows the definitions of pulse and slope duration.

reasoning as done for equation (3.7) and calculating the new eigenenergies of the system from theHamiltonian

Hstark =~δ2

σz + ~Ωσx. (3.13)

Here only carrier transitions were taken into account. For δ >> Ω one finds that the upper levelwill be shifted by Ω2/δ and the lower level will be shifted by −Ω2/δ. This amounts to a totalfrequency shift of the transition by

δstark = 2Ω2

δ. (3.14)

From the Hamiltonian one can already see that such an interaction is proportional to a σz rotationon a qubit. Thus by shining in a detuned light field for a certain amount of time one can performan arbitrary rotation around the z-axes realizing the third single qubit operation mentioned inchapter 2.1.2.

So far it seems that Stark shifts are only desired interactions. This changes when one con-siders exciting sideband transitions like |↓, 0〉 → |↑, 1〉 which are needed for some two-qubit gateimplementations [14]. These gate-operations are similar to interference experiments, thus it is ofutmost importance that the phase relation between pulses does not vary over time. Due to theAC-Stark shift generated by the carrier, intensity fluctuations would be immediately transferredinto phase fluctuations and thus spoil the quality of the gate operation. A possibility to get rid ofthese amplitude-dependent Stark shifts is to shine in a second light field detuned in opposite direc-tion with a Rabi frequency exactly canceling the effect. This light field is derived from the samebeam as the driving field and thus has the same amplitude fluctuations which together leads to acommon-mode rejection of the Stark shift. A thorough explanation and analysis of this techniquecan be found in [93, 103].

3.2.2. Off resonant excitation

A light field resonant with the transition |↓, 0〉 → |↑, 1〉 will not only drive Rabi oscillations onthis transition but also off-resonantly drive the carrier transition |↓, 0〉 → |↑, 0〉. This effect can beunderstood by looking at the time evolution of the |D, 0〉 population, which is obtained by solving

22

3.3 The S1/2 → D5/2 transition

the optical Bloch equations [104]

P|↑,0〉 =(2Ω)2

(2Ω)2 + ∆2sin2

(12

√(2Ω)2 + ∆2 t

). (3.15)

Typical experimental parameters are Ω = 2π · 45 kHz and an axial trap frequency and thusdetuning of ∆ = ωax = 2π ·1.2 MHz. This will result in off-resonant oscillations with an amplitudeof 2.5% and a frequency of 1.2 MHz. Rabi oscillation on the carrier and the sideband are shown infigure 3.2(a). As gate operations have to reach fidelities exceeding 99% one has to find a methodto avoid this effect.

The method of choice is to switch pulses adiabatically as shown in figure 3.2(b). This pulseshaping has two effects. First, it suppresses high frequency components in the Fourier transform ofthe pulse which are related to the off-resonant excitations. Second, due to the adiabatic switchingwhich starts and stops with a zero amplitude of the pulse, the off-resonant excitations are adia-batically transformed to zero. Typical pulse slopes are 5 to 10 µs for a light power equivalent toperforming a full rabi oscillation within 100 µs on the blue sideband. Further information on pulseshaping and off-resonant excitations can be found in [105].

3.3. The S1/2 → D5/2 transition

For all but one experiment in this thesis the dipole forbidden transition S1/2 → D5/2 of 40Ca+

and 43Ca+ was used to realize a qubit. In this section the excitation of the quadrupole transitionS1/2 ↔ D5/2 by laser light is discussed which is important to understand the experiments presentedin chapters 6, 7 and 8.

3.3.1. Rabi frequency and geometrical considerations

For a quadrupole transition, the induced electric quadrupole moment Q couples to the gradient ofthe applied electric field E. The Hamiltonian describing this interaction is given by

HI = −Q · ∇E(t). (3.16)

This type of interaction takes on the shape of equation (3.3) when the Rabi frequency is definedas

Ω =∣∣∣∣eE0

4~

⟨S1/2, F,mF |(~ε · ~r)(~k · ~r)|D5/2, F

′,m′F

⟩∣∣∣∣ (3.17)

with E0 the electric field amplitude and ~r the position operator of the valence electron relativeto the atomic nucleus. For 40Ca+ the quantum numbers F, F ′ can be ignored and mF ,m′

F bereplaced by mj ,m

′j . The selection rules allow for ∆m = mj − m′

j = 0,±1,±2. In the case of43Ca+, the same rules apply for mF ,m′

F but additionally ∆F = F − F ′ = 0,±1,±2 has to befulfilled. From the term (~ε · ~r)(~k · ~r) one can already guess that the effective coupling strengthdepends on the laser beam direction and polarization relative to the quantization axis defined by

23


the magnetic field vector. The expression for the Rabi frequency can be rewritten in the form [94]

Ω =eE0

4~

√15cα

ΓD5/2

k3Λ(F,m, F ′, m′) g∆m(φ, γ) (3.18)

with the fine structure constant α, the speed of light c, the electron charge e and the spontaneousdecay rate ΓD5/2 . The coupling strength Λ(F, m,F ′,m′) contains all properties of the atomictransition apart from geometrical factors. The coefficients for all transitions in 40Ca+ and 43Ca+

are listed in Appendix C.The function g∆m(φ, γ) contains the geometry dependence of Ω [92, 98] where φ denotes the

angle between the laser beam and the magnetic field and γ describes the angle between a linearpolarization and the magnetic field vector projected onto the plain of incidence. Choosing ~B =B0(0, 0, 1), one obtains for ~k = k(sinφ, 0, cos φ)1 and ~ε = (cos γ cosφ, sin γ,− cos γ sin φ)

g(0) =12| cos γ sin(2φ)| (3.19)

g(±1) =1√6| cos γ cos(2φ) + i sin γ cos φ| (3.20)

g(±2) =1√6|12

cos γ sin(2φ) + i sin γ sin φ| (3.21)

These terms change for circularly polarized light with ~ε = 1√2(p · i cos φ, 1,−p · i sin φ) and p = ±1

for σ± polarized light to

g(0) =1√8| sin(2φ)| (3.22)

g(±1) =1√12

| cos(φ)−∆m · p · cos(2φ)| (3.23)

g(±2) =1√12

|12

sin(2φ) + sin φ| (3.24)

Since any Zeeman transition from S1/2 → D5/2 can be used for coherent manipulation of the ions’quantum state, it is interesting to look at the possibility to change the coupling strengths of thesetransitions. Three cases are especially useful when it comes to maximizing the coupling strengthof a particular transition and minimize others:

• Linear polarized light with φ = 90, γ = 90. Exciting only ∆m = +2 and ∆m = −2transitions. This configuration could be used for sideband cooling (see chapter 5).

• Linear polarized light with φ = 45, γ = 0. Here the ∆m = 0 transitions couplestrongest to the laser while ∆m = ±1 are completely suppressed. In this configuration onewould use e.g. the 40Ca+ S1/2(m = +1/2) → D5/2(m = +1/2) transition for storing thequbit while doing the sideband cooling on S1/2(m = +1/2) → D5/2(m = +5/2).

• Circular polarized light along the magnetic field axis φ = 0. This is the onlyconfiguration where a single transition can be selected by using σ± polarized light to excite∆m = ±1 transitions. This also results in the biggest coupling strength that can be achieved.

1the y-component can be chosen to be zero due to cylindrical symmetry

24

3.4 Microwave transitions

A disadvantage of this configuration is that sideband cooling is not possible as the coolingcycle can only be closed via the P3/2(mj = +3/2) state when the S1/2(mj = +1/2) →D5/2(mj = +5/2) transition is excited.

Configuration 3 is especially interesting in 43Ca+ to reduce the number of possible transitions bya considerable amount. Thereby one can also suppress unwanted off-resonant excitation and Starkshifts due to other carrier transitions close by. A more mathematical treatment of this subjectwith equations that are valid for arbitrary polarization and angle can be found in [92].

3.4. Microwave transitions

For the hyperfine ground state manifold of 43Ca+ microwave transitions are of importance tomanipulate the qubit. A microwave field couples to the magnetic dipole moment of the ion. TheRabi frequency can be expressed as

ΩMW =∣∣∣∣

12~

⟨S1/2, F = 4,mF

∣∣ ~µB~BMW

∣∣S1/2, F′ = 3,m′

F

⟩∣∣∣∣ (3.25)

with ~µB the ion’s magnetic dipole moment and ~BMW the magnetic field of the microwave radiation.Similar to dipole transitions one can drive a ∆mF = 0 transition with a π-polarized AC magneticfield, whereas for ∆mF = ±1 transitions σ±-polarized fields are needed.

With microwave fields it is impossible to directly drive sideband transitions as the Lamb-Dickeparameter is extremely small. For trapping frequencies of 1 MHz and the ground state hyperfinesplitting in 43Ca+ of 3.2 GHz the Lamb-Dicke parameter is on the order of 10−6. Another drawbackis that microwave fields can not be used to address single ions. The wavelength is so big that nofocussing can be achieved with respect to the few µm distance of the ions. Both problems mightbe circumvented by either using high magnetic field gradients [106] and/or oscillating magneticfields [107]. These approaches are quite challenging and were not pursued in this thesis.

Although microwave transitions have drawbacks they are a valuable tool for reliably driving thehyperfine qubit, especially if all ions in the trap are subject to the same operation. In this thesisthe microwave field was used to map between the hyperfine and the optical qubit (see chapter 6)and to characterize the coherence properties of the hyperfine qubit.

3.5. Mølmer-Sørensen gate

A two-qubit quantum gate-operation that is equivalent to a controlled-NOT gate up to localoperations is achieved by the action of a Hamiltonian H ∝ σn⊗σn, where σn = σ · n is a projectionof the vector of Pauli spin matrices onto the direction n [108]. Two prominent examples of this typeof gate are the conditional phase gate [109, 110] and the Mølmer-Sørensen gate [34, 80, 111] (MS-gate). In the latter case, correlated spin flips between the states |↑〉 |↑〉 ↔ |↓〉 |↓〉 and |↑〉 |↓〉 ↔ |↓〉 |↑〉are induced by a Hamiltonian

H ∝ σφ⊗σφ where σφ = cos φ σx + sin φ σy. (3.26)

25


Figure 3.3.: Mølmer-Sørensen interaction scheme. A bichromatic laser field couples thequbit states |↓↓〉 ↔ |↑↑〉 via the four interfering paths shown in the figure. Similar processescouple the states |↑↓〉 ↔ |↓↑〉. The frequencies ν± of the laser field are tuned close to thered and the blue motional sidebands of the qubit transition with frequency ν0, and satisfythe resonance condition 2ν0 = ν+ + ν−. The vibrational quantum number is denoted n.

The unitary operation UMSφ = exp(iπ

4 σφ ⊗ σφ) maps product states onto maximally entangledstates. In 1999 the proposal was made to realize an effective Hamiltonian [80, 111] taking theform (3.26) by exciting both ions simultaneously with a bichromatic laser beam with frequenciesν± = ν0±δ where ν0 is the qubit transition frequency and δ is close to a vibrational mode of the two-ion crystal with frequency ω. Figure 3.3 shows the level scheme of two ions in a harmonic oscillatorand the laser fields applied for a Mølmer-Sørensen interaction. Changing into an interaction pictureand performing a rotating-wave approximation, the time-dependent Hamiltonian

H(t) = ~Ω(e−iδt + eiδt)eiη(ae−iωt+a†eiωt)(σ(1)+ + σ

(2)+ ) + h.c. (3.27)

is well approximated byH(t) = −~ηΩ(a†ei(ω−δ)t + ae−i(ω−δ)t)Sy (3.28)

in the Lamb-Dicke regime.

In equation (3.28), we use a collective spin operator Sy = σ(1)y +σ

(2)y . In the following we denote

the laser detuning from the motional sidebands by ω − δ = ε. The whole subsequent treatmentof the MS-interaction can be generalized for N ions by exchanging Sy with SN

y =∑N

i=0 σ(i)y . The

Hamiltonian (3.28) can be exactly integrated [81] yielding the propagator

U(t) = D(α(t)Sy)e(i(λt−χ sin(εt))S2y), (3.29)

where α(t) = ηΩε (eiεt − 1), λ = η2Ω2/ε, χ = η2Ω2/ε2, and D(α) = eαa†−α∗a is a displacement

operator. For a gate time tgate = 2π/|ε|, the displacement operator vanishes so that the propagatorU(tgate) = exp(iλtgateS

2y) can be regarded as being the action of an effective Hamiltonian

Heff = −~λS2y = −2~λ(1 + σy ⊗ σy) (3.30)

inducing the same action up to a global phase as the one given in (3.26). Setting Ω = |ε|/(4η), agate is realized, which is capable of maximally entangling ions irrespective of their motional state.

In the description of the gate mechanism given so far, a coupling of the light field to the carriertransition was neglected based on the assumption that the Rabi frequency Ω was small compared

26

3.5 Mølmer-Sørensen gate

gate operation gate operation

z

yx

z

yx

z p= /2z = 0(a) (b)

Figure 3.4.: Effect of non-resonant excitation of the carrier transition. (a) For ζ = 0,the gate starts at a maximum of the intensity-modulated beam. In this case, a Blochvector initially centered at the south pole of the Bloch sphere performs oscillations that aresymmetric around the initial position. (b) For ζ = π/2, the gate starts at the minimum ofthe intensity modulation. In this case, the average orientation of the Bloch vector is tiltedwith respect to its initial position.

to the detuning δ of the laser frequency components from the transition. In this case, small non-resonant Rabi oscillations that appear on top of the gate dynamics are the main effect of coupling tothe carrier transition. Since a maximally entangling gate requires a Rabi frequency Ω ∝ η−1t−1

gate,the question of whether Ω ¿ δ holds becomes crucial in the limit of fast gate operations andsmall Lamb-Dicke factors. Our experiments [40] are exactly operating in this regime, and it turnsout that non-resonant excitation of the carrier transition has further effects beyond inducing non-resonant oscillations [112]. This becomes apparent by interpreting terms in the Hamiltonian in adifferent way: The red- and blue-detuned frequency components E± = E0 cos((ν0±δ)t±ζ) of equalintensity may be viewed as a single laser beam E(t) = E+ + E− = 2E0 cos(ν0t) cos(δt + ζ) that isresonant with the qubit transition but amplitude-modulated with frequency δ. Here, the phase ζ

which determines whether the gate operation starts in a maximum (ζ = 0) or a minimum (ζ = π/2)of the intensity of the amplitude-modulated beam has a crucial influence on the gate-operation.This can be intuitively understood by considering the initial action the gate exerts on an inputstate in the Bloch sphere picture shown in figure 3.4. For short times, coupling to the sidebandscan be neglected which justifies the use of a single-ion picture. The dynamics is essentially theone of two uncoupled qubits. The fast dynamics of the gate is induced by excitation of the ionson the carrier transition. For ζ = 0, the Bloch vector of an ion initially in state |↓〉 will oscillatewith frequency δ along a line centered on the south pole of the Bloch sphere. For ζ = π/2, theoscillation frequency is the same, however, the time-averaged position of the Bloch vector is tiltedby an angle

ψ =4Ωδ

sin ζ (3.31)

with respect to the initial state |↓〉. This effect has a profound influence on the gate action.A careful analysis of the gate mechanism [112] taking into account the non-resonant oscillationsreveals that the Hamiltonian (3.28) is changed into

H(t) = −~ηΩ(a†ei(ω−δ)t + ae−i(ω−δ)t)Sy,ψ , (3.32)

27


whereSy,ψ = Sy cosψ + Sz sin ψ , (3.33)

and that the propagator (3.29) needs to be replaced by

U(t) = e(−iF (t)Sx)D(α(t)Sy,ψ)e(i(λt− χ sin(εt))S2

y,ψ

), (3.34)

where the term containing F (t) = 2Ωδ (sin(δt + ζ) − sin ζ) describes non-resonant excitation of

the carrier transition. In the derivation of Hamiltonian (3.32), small terms arising from the non-commutativity of the operators Sy, Sz have been neglected [112]. The dependence of the propagatoron the exact value of ζ is inconvenient from an experimental point of view. To realize the desiredgate-operation in an optimal way, precise control over ζ is required. In addition, the gate durationmust be controlled to better than a fraction of the mode oscillation period because of the non-resonant oscillation. Fortunately, both of these problems can be overcome by shaping the overallintensity of the laser pulse such that the Rabi frequency Ω(t) smoothly rises within a few cycles 2π/δ

to its maximum value Ωgate ≈ |ε|/(4η) and smoothly falls off to zero at the end of the gate. In thiscase, the non-resonant oscillations vanish and (3.31) shows that the operator Sy,ψ(t) adiabaticallyfollows the laser intensity so that it starts and ends as the desired operator Sy irrespective ofthe phase ζ. For intensity-shaped pulses, the propagator (3.29) provides therefore an adequatedescription of the gate action.

3.5.1. AC-Stark shifts

In the description of the gate mechanism given so far the ion was treated as an ideal two-levelsystem. AC-Stark shifts are completely insignificant provided that the intensities of the blue- andthe red-detuned frequency components are the same since in this case light shifts of the carriertransition caused by the blue-detuned part are exactly canceled by light shifts of the red-detunedlight field. Similarly, light shifts of the blue-detuned frequency component non-resonantly excitingthe upper motional sideband are perfectly canceled by light shifts of the red-detuned frequencycomponent coupling to the lower motional sideband.

For an experimental implementation with calcium ions, we need to consider numerous energylevels (see figure 2.3). Here, the laser field inducing the gate action causes AC-Stark shifts on thequbit transition frequency due to non-resonant excitation of far-detuned dipole transitions and alsoof other S1/2 ↔ D5/2 Zeeman transitions. The main contributions arise from couplings betweenthe qubit states and the 4p−states that are mediated by the dipole transitions S1/2 ↔ P1/2,S1/2 ↔ P3/2, D5/2 ↔ P3/2. Other transitions hardly matter as can be checked by comparing theexperimental results obtained in [103] with numerical results based on the transition strengths [98]of the dipole transitions coupling to the 4p−states. For suitably chosen k-vector and polarizationof the bichromatic laser beam, these shifts are considerably smaller than the strength λ of the gateinteraction.

AC-Stark shifts can be compensated by a suitable detuning of the gate laser. An alternativestrategy consists in introducing an additional AC-Stark shift of opposite sign that is also caused bythe gate laser beam [103]. This approach has the advantage of making the AC-Stark compensationindependent of the gate laser intensity. In contrast to previous gate-operations relying on this

28

3.5 Mølmer-Sørensen gate

technique [113] where the AC-Stark shift was caused by the quadrupole transition and compensatedby coupling to dipole transitions, here, the AC-Stark shift is due to dipole transitions and needsto be compensated by coupling to the quadrupole transition.

For ions prepared in the ground state of motion (n = 0), a convenient way of accomplishing thistask is to perform the gate operation with slightly imbalanced intensities of the blue- and the red-detuned laser frequency components. Setting the Rabi frequency of the blue-detuned componentto Ωb = Ω(1 + ξ) and the one of the red-detuned to Ωr = Ω(1− ξ), an additional light shift causedby coupling to the carrier transition is induced that amounts to δ

(C)ac = 2(Ω2

r −Ω2b)/δ = −8Ω2ξ/δ.

Now, the beam imbalance parameter ξ needs to be set such that the additional light shift exactlycancels the phase shift φ = δactgate induced by the dipole transitions during the action of the gate.Taking into account that tgate = 2π/ε and Ω = |ε|/(4η), this requires ξ = (δη2/|ε|)(φ/π).

Apart from introducing light shifts, setting ξ 6= 0 also slightly changes the gate Hamiltonian(3.30) from Heff = −λS2

y to Heff = −λ(S2y + ξ2S2

x) [114] since the coupling between the states |↓↓〉,|↑↑〉 is proportional to 2ΩbΩr = 2Ω2(1−ξ2) whereas the coupling between |↓↑〉, |↑↓〉 is proportionalto Ω2

b + Ω2r = 2Ω2(1 + ξ2). However, as long as ξ ¿ 1 holds – which is the case in the experiments

described in chapter 6 – this effect is extremely small2 as the additional term is only quadratic inξ.

Another side effect of setting ξ 6= 0 is the occurrence of an additional term ∝ Sza†a in the

Hamiltonian. It is caused by AC-Stark shifts arising from coupling to the upper and lower motionalsideband which no longer completely cancel each other. The resulting shift of the qubit transitionfrequency depends on the vibrational quantum number n and is given by δ(SB) = (8η2Ω2/ε)ξn =(ε/2)ξn. Simulations of the gate action based on (3.27) including an additional term ∝ Sz ac-counting for AC-Stark shifts of the dipole transitions and power-imbalanced beams show that theunwanted term ∝ Sza

†a has no severe effects for ions prepared in the motional ground state as longas ξ ¿ 1. However, for ions in Fock states with n > 0, this is not the case. Taking the parameterset ξ = 0.075, ω = (2π) 1230 kHz, and ε = (2π) 20 kHz as an example, the following numericalresults are obtained: applying the gate to ions prepared in | ↓↓〉|n = 0〉, a Bell state is createdwith fidelity 0.9993. For n = 1, the fidelity drops to 0.985, and for n = 2 to even 0.942. This lossof fidelity can be only partially recovered by shifting the laser frequency by δ(SB), the resultingfidelity being 0.993 and 0.968, for n = 1 and n = 2 respectively. For higher motional states, theeffect is even more severe and shows that this kind of AC-Stark compensation is inappropriatewhen dealing with ions in a thermal state of motion with n À 1. Instead of compensating theAC-Stark shift by imbalancing the beam powers, in this case, the laser frequency needs to beadjusted accordingly (see chapter 6 on the experiments with Doppler-cooled ions).

3.5.2. A Mølmer-Sørensen interaction with ions in a thermal state of motion

In theory, the Mølmer-Sørensen gate does not require the ions to be cooled to the ground statesof motion since its propagator (3.29) is independent of the vibrational state for t = tgate. For

2The additional term S2x changes the gate operation from U = exp (−i π

8S2

y) to Uξ = exp (−i π8(S2

y + ξ2S2x)). A

short calculation shows that the minimum state fidelity Fmin = minψ(|〈ψ|U†Uξ|ψ〉|2) is given by Fmin =12(1 + cos(π

2ξ2)) where we used [S2

x, S2y ] = 0 and exp(−iγS2

x) = 1 + 14(e−i4γ − 1)S2

X . Thus, for ξ = 0.075, one

obtains Fmin ≈ 1− 2 · 10−5.

29


t 6= tgate, however, the interaction entangles qubit states and vibrational states so that the qubits’final state becomes dependent on the initial vibrational state. Therefore, it is of interest to calculateexpectation values of observables acting on the qubit state space after applying the propagator foran arbitrary time t. Simple expressions can be derived [76] in the case of a thermally occupiedmotional state where the mode population probabilities are given by

pn =1

n + 1

(n

n + 1

)n

. (3.35)

Not taking into account non-resonant carrier oscillations, one finds the following expressions forthe qubit state populations

p↑↑(t) =18(3 + e−16|α|2(n+ 1

2 ) + 4 cos(4γ)e−4|α|2(n+ 12 ))

p↑↓(t) + p↓↑(t) =14(1− e−16|α|2(n+ 1

2 )) (3.36)

p↓↓(t) =18(3 + e−16|α|2(n+ 1

2 ) − 4 cos(4γ)e−4|α|2(n+ 12 ))

with α = α(t) and γ = λt− χ sin(εt) containing the time dependent terms.

3.6. Creating a coherent displacement

Another application for bichromatic light fields is to create coherent displacement forces. This canbe done by shining in two light fields resonant with the blue and red sideband transition. Puttingtogether equations (3.12) and (3.11) and calculating a few lines to simplify the expressions onearrives at the following Hamiltonian

HD = ~ηΩ [σx cosφ+ − σy sin φ+]

⊗ [(a + a†) cos φ− + i(a† − a) sin φ−

]. (3.37)

Here 2φ+ = φr + φb and 2φ− = φb − φr are the sum and the difference, respectively, of the phasesof the light fields tuned to the red and blue sideband. One can recognize the position x andmomentum operators p in the second part of the Hamiltonian. The propagator for φ− = φ+ = 0is then

UD = e−iηΩt

∆ xσx (3.38)

which describes a state dependent coherent displacement operator D(ασx) with α = iηΩt/∆ (seechapter 2). The propagator (3.38) displaces an eigenstate of σx along p in phase space. If theinitial state was in the ground state a coherent state is created. By properly choosing the phasesφ−, φ+ one can create coherent states with arbitrary α.

3.6.1. Measurement of 〈x〉 and 〈p〉A lot of ion trap experiments make extensive use of the harmonic oscillator created by the trapconfinement for implementing two-qubit interactions. The ion trap system is always referred to as

30

3.6 Creating a coherent displacement

“the“ toy model for a quantum harmonic oscillator. Nevertheless the expectation values for theposition and momentum operators 〈x〉 and 〈p〉 have never been measured with trapped ions up tonow. Here a method is presented similar to one proposed in [115–117] to measure these quantitieswithout reconstructing the full quantum state.

As already mentioned in chapter 2.1.3 the only observable that can be directly measured in iontrap experiments by fluorescence detection is σz. By using additional laser pulses one can generatea propagator U that maps other observables onto σz. In this case we apply the propagator UD tothe state ρ of the ion prior to the state detection which is equivalent to measuring the observable

A(t) = U†DσzUD = cos(2ηxΩpt/∆)σz + sin(2ηxΩpt/∆)σy (3.39)

on the initial state because of Tr((U†ρU)σz) = Tr(ρ(UσzU†)). If the ion’s internal initial state is

the eigenstate of σz belonging to eigenvalue +1, 〈A(t)〉 = cos(2ηxΩpt/∆) and for the eigenstate ofσy belonging to eigenvalue +1, 〈A(t)〉 = sin(2ηxΩpt/∆).

Thus preparing ρ in an eigenstate of σy and taking the first derivative of the measured expectationvalue

d

dt〈A(t)〉

∣∣∣∣t=0

= 2ηΩp/∆〈xσy〉 (3.40)

reveals that the slope of the measured excitation at t = 0 is proportional to 〈x〉. By changing thephase φ− from 0 to π/2 it is possible to measure 〈p〉 instead of 〈x〉. Furthermore all n-momenta ofx can be measured by taking the n-th derivative

dn

dtn〈A(t)〉

∣∣∣∣t=0

∝ 〈xnσy〉. (3.41)

We have applied this technique to determine position and momentum in a simulation of a Diracparticle encoded in position and momentum of the harmonic oscillator (see chapter 8).

31


32

4. Experimental setup

In this chapter an experimental setup is described used to realize the experiments discussed in theproceeding chapters. A section about the ion trap and the vacuum setup will be followed by adescription of the laser system and the computer control.

4.1. Linear ion trap and electrode wiring

The ion trap used in this thesis is the standard Innsbruck design (see figure 4.1(a)). It is alinear Paul trap where the radio frequency (RF) drive is applied to one of two pairs of bladeswhile the other pair is grounded. This creates the radial confinement. An additional DC voltageapplied to the tips generates the axial confinement (see figure 4.1(b)). A thorough discussion ofthe specifications and design of the trap can be found in Jan Benhelm’s thesis [94].

The ion trap is operated at a frequency of about 25.5 MHz produced by a signal generator1 whoseoutput is amplified2 to 5-13 W. This signal is then coupled into a helical resonator amplifying thevoltage to about 1200 V which is applied to the blades. This creates a harmonic potential withtrapping frequencies between 2.5 MHz and 4 MHz in the radial directions. The two radial modesare almost degenerate in frequency and measurements showed that they are split by about 50 kHz.The orientation of these modes is along the RF-blades of the trap. This breaking of the radialsymmetry is due to the fact that the RF potential is only applied to one blade pair while the otheris grounded. The DC voltage applied to the tips ranges between 800-1400 V. A typical voltage of1200 V creates an axial potential with a trap frequency of 1.36 MHz for a 40Ca+ ion.1Rohde & Schwarz, SML012Mini Circuits LZY-1

Figure 4.1.: (a) Picture of the linear Paul trap used in the experiment. (b) Schematicdrawing of the trap. Two opposite blade shaped electrodes (A) are connected to a highvoltage radio frequency while the other pair is grounded. The tips (B) are connected to aDC voltage up to 1400 V. The compensation electrodes (C) are used to shift the ions intothe RF null compensating stray electric fields. The upper compensation electrode of thepair used for horizontal micromotion compensation is also used to guide microwave signalsto the ions. The distance of the tips and electrodes is given in mm.

33

Experimental setup

Figure 4.2.: Wiring of the DC-trap electrodes. Each tip is wired to a circuit that canchange the voltages on the tips by twice the preset voltages U1/U2. A TTL signal switchesbetween the two voltages and the RC combination is designed such that voltage reaches thedesired value within 40 µs. This time constant is given by the low pass filters attached tothe tips. The virtual ground Ug for the secondary side is provided by a stable high voltagesource (ISEG). The effective voltage on the tips is then given by Utip1 = Ug ± U1/2 andUtip2 = Ug ∓ U1/2.

These voltage values lead to a string of ions in the trap along the direction of the tips. Theequilibrium positions of the ions are determined by the external trapping potential and the Coulombrepulsion between the ions. For two ions of mass m the distance can be calculated analytically andis given by

∆z =(

e2

2πε0mωz

)1/3

. (4.1)

For an axial center of mass frequency ωz =1.2 MHz this leads to an ion spacing of 4.6 µm. Thisknowledge is used to calibrate the imaging system in combination with the precisely measurabletrap frequency.

Each tip is wired separately such that it is possible to put them on different potentials. Bychanging the potentials applied to the tips the ion string can be moved along the trap axis. Toshuttle the ions fast without compromising the stability of the trap potential we designed a circuitwhich can change the tip voltage by ±20 V. With a TTL signal one can switch between two presetvalues U1, U2 such that the voltage on the tips changes within 40 µs. The time constant is givenby the low pass filters attached to the tips. On the one hand these filters block noise going tothe trap electrodes and on the other hand they shield the electronics from RF pick up by thetips. A schematic drawing of the circuit can be seen in figure 4.2. A stable high voltage3 sourceprovides the virtual ground (Ug = 800−1200 V ) for the secondary side of the analog optocoupler4.A power supply referenced to the same high voltage provides the necessary current to drive theadder/multplier circuit. This design allows one to combine the stability of the high voltage box< 10−5 with a fast switching capability. The instability of the voltages U1, U2 (∆V/V ≈ 10−3)and the noise of the electronic circuit (∆V/V ≈ 10−3) do not harm. This comes from the fact,that the noise has to be compared to the 800 to 1200 V provided by the high voltage source. Theeffective noise is less than 10−5 which is sufficient for all experiments in this thesis. The effectivevoltage on the tips is given by Utip1 = Ug ± U1/2 and Utip2 = Ug ∓ U1/2.

Due to electric stray fields, the ion position can be pushed out of the RF null leading to a motionof the ion with a frequency which is the same as the drive frequency. This excess micromotion

3ISEG, EHQ F020p, 0-2kV, ∆V/V < 10−5

4HCNR201 Agilent Tech.

34

4.2 Vacuum vessel

Figure 4.3.: (a) Schematic drawing of the trap setup viewed from atop showing traporientation, laser beam directions and positioning of the inverted viewports. Most of thelaser beams entering the chamber are within the equatorial plane. Two coils in almostHelmholtz configuration generate a magnetic field defining the quantization axes along SW-NE. An additional coil pair along SE-NW is used for compensating external magnetic strayfields. The inverted view ports allow us to place custom-made lenses close to the ions. Thelenses are used to focus laser beams tightly and at the same time collect a big fraction ofthe fluorescence light emitted by the ions. (b) Side view of the setup. The viewports at thetop and the bottom of the vacuum chamber are mounted under an angle of 60 with respectto the equatorial plane. Additional beams for Doppler cooling and ion state manipulationare guided through these windows. The single coil on top of the vacuum chamber is used tocompensate the earth’s magnetic field.

can be minimized by compensating for the stray electric fields and shifting the ion back into thepotential node. This can be done by applying small voltages to the compensation electrodes,ranging typically from 1 V to 150 V.

4.2. Vacuum vessel

The vacuum chamber housing the ion trap is a stainless steel octagon with eight CF63 flanges.Three of the eight flanges are equipped with inverted viewports5 the rest have regular viewportsattached6. The inverted viewports allow us to place custom made lenses7 close to the ions and tosteer or replace them without opening the vacuum chamber. The proximity to the ions guarantees agood imaging resolution, a high photon collection efficiency and the ability to tightly focus lasers toaddress individual ions. The top and the bottom of the chamber consist of two CF200 flanges eachcarrying two CF40 flanges with windows mounted under an angle of 60 relative to the equatorialplane. Additional CF16 feedthroughs are attached to the bottom and top flange for wiring the iontrap and the calcium oven. A schematic drawing can be seen in figure 4.3a.

Not shown in this drawing are the ion getter pump8, the titanium sublimation pump9 andthe Bayard-Alpert-Gauge10 attached to the chamber via a six way cross. The ion pump runs

5Ukaea, fused silica, anti reflection coating (Tafelmaier, 397nm and 720-870nm on vacuum side only)6Ukaea, fused silica, anti reflection coating (Tafelmaier, 397nm and 720-870nm)7Silloptics, Germany8Varian Star Cell, 20 l9Varian

10Varian, UHV-24 Gauge

35

Experimental setup

permanently while the titanium sublimation pump is fired once a week. The pressure in thechamber stays below the measurement limit of the Bayard-Alpert gauge of 2 · 10−11 mbar.

4.3. Magnetic field coils

Two pairs of magnetic field coils are attached to the vacuum chamber (see figure 4.3a). The coilpair aligned along the SW-NE direction is generating the quantization field. The distance of thecoils is 300 mm while their inner diameter is 115 mm. This mounting does not quite resemble aHelmholtz configuration but nevertheless we expect only small magnetic field gradients at the ionposition. Measurements with two entangled ions confirmed that the gradient at the trap center issmaller than 0.2 G/m. This means that the frequency change, due to this gradient, is smaller than1 Hz on the qubit transition for two ions separated by 4 µm. The coils are made out of 300 windingsof copper wire. A current of 1 A sent through these coils produces a magnetic field of 3.4 G at theion position. The second pair aligned along the SE-NW11 direction is for compensating externalmagnetic fields in order to align the quantization axis with the SW-NE direction. An additionalsingle coil with a 200 mm diameter (see figure 4.3b) is mounted on the top flange to compensatefor magnetic fields in the vertical direction.

All coils are powered by home-built current drivers having a relative current drift of less than2 · 10−5 in 24 h. This is achieved by actively stabilizing the current by a servo loop. A stablereference voltage is compared with the voltage drop across a highly stable resistor12 and used as afeedback signal.

4.4. Optical access and single ion addressing

Several laser beams enter the vacuum chamber from seven different directions (see figure 4.3).A 397 nm σ+-polarized beam is aligned exactly with the magnetic field direction. It is used foroptical pumping of 40Ca+ and 43Ca+ and as an additional Doppler cooling beam for 43Ca+. Alllaser beams for Doppler cooling, repumping and photoionization are sent through the SE port ofthe chamber. The beams are focused by adjusting the output coupler of the fibers. Typical beamwaists are on the order of 100-200 µm. Additional beams for Doppler cooling and repumpingare sent in via the viewports in the bottom of the vessel. These beams are used when a secondphotomultiplier tube (PMT)13 at the NW port is added to the primary PMT located at the southviewport. The only beam that remains operational from the SE corner is the photoionizationbeam. In order not to damage the second PMT during loading the light is automatically blockedby a shutter in front of the PMT whenever the photoionization lasers are on.

Four 729 nm beams for manipulating the qubits are sent onto the ions. The one entering fromthe inverted viewport S is used for addressing a single ion out of a chain of ions. This is achievedby a custom-made lens system14 and additional telescope lenses to widen the beam prior to thefocussing (see figure 4.4(a)). With this setup the expected diffraction limited spot size is 2.9 µm.

11all directions are given according to the compass card in figure 4.312Vishay, VCS 30213Electron tubes, P25C, Quantum efficiency = 28% at 400 nm14Sill Optics, Germany, antireflection coated for 397 nm and 729 nm

36

4.5 Fluorescence detection

Figure 4.4.: (a) Custom-made lens setup for focusing a 729 nm laser down to < 4 µmfor addressing a single ion. The same lens system together with a dichroic mirror is used toimage the ion string onto the PMT and the camera. The objective covers about 2.5 % ofthe full solid angle. A dichroic mirror is used to separate the incoming light at 729 nm fromthe fluorescence light at 397 nm. (b) Measurement of the intensity profile of the laser beamat 729 nm entering from the south port of the vacuum chamber. For this measurement twoions where first shifted in position with respect to the laser beam and then excited by alaser pulse. The pulse length was chosen such that the ion undergoes 3π rabi flops when thebeam is centered on it. The intensity of the beam was then calculated with the recordedexcitation.

Two ions were used to probe the experimentally achieved width of the beam by moving themalong the trap axis and exciting them with a laser beam (see figure 4.4(b)). The pulse length andintensity of the laser pulse were chosen such that the ion undergoes a 3π rotation when the beamis centered on it. The spatial intensity profile of the beam was then deduced from the recordedexcitation to the D state. This measurement revealed a FWHM of the beam of 2.6 µm.

Two beams coming from the directions NE and NW are focussed to about 20 µm. The NEbeam is σ+ polarized for driving the S1/2, mF = 1/2 to D5/2,mF = 3/2 transition with maximalefficiency. The NW beam is linearly polarized for driving ∆m = ±2 transitions with maximalefficiency. The beam size is a compromise between maximum intensity and the requirement toilluminate up to three ions equally. Furthermore, the ions should not reside at a position wherethe intensity profile has a step gradient to minimize coupling strength fluctuations due to beamsteering effects. The two beams are not operated simultaneously. In the experiment we can switchfrom one to the other by exchanging the fiber after the switching AOM (see figure 4.7 AO3).Another 729 nm beam enters the vacuum chamber from a viewport in the bottom of the chamberunder an angle of 60 with respect to the equatorial plane. This beam is only focused with theoutput coupler of the fiber to about 150 µm. It is used to illuminate all trapped ions equally.

4.5. Fluorescence detection

To collect the fluorescence light at 397 nm we again utilize the custom made lens design used forfocusing the beams at 729 nm (see figure 4.4(a)). It is a five-lens objective which is antireflectioncoated for 397 nm and 729 nm. The abberations introduced by the fused silica window of theinverted viewport are corrected by the lens-design. The entrance diameter of the objective isd=38 mm and the distance to the ions from the first lens is r=58 mm. The photon collection

37

Experimental setup

Figure 4.5.: (a) Fluorescence histogram for two ions each in a superposition of S and D.In this plot laser intensity and detection time (3 ms) where chosen such that a detectionfidelity exceeding 99.9% was achieved. Two thresholds are used to discriminate betweenthree different two ion states. Either both ions are in the D state, both are in the S stateor one is in the D state and the other in the S state with a respective mean photon countrate of 30, 270, 510 photons in 3 ms. The probabilities of finding k bright ions are denotedpk. (b) The histogram illustrates the discrimination of the two-qubit states S1/2 and D5/2

for a single ion using two PMTs. For each data point the fluorescence of the ion, preparedin a superposition of the S and D state, was measured for 250 µs. On average we acquire0.07 photons during this detection time for the ion in the D state. If the ion is in the Sstate 7.8 photons are detected. By introducing a threshold between one and two detectedphotons one can discriminate between the two states. The intensity of the detection laserwas chosen such that the ion was not heated beyond the mean phonon number achievedwith Doppler cooling. Additionally the time of the detection was made as short as possibleto shorten the duration of the sequence. Despite these restrictions a detection infidelityof wrongly assigning a quantum state of 0.32 % was achieved. For an application of thisdetection technique see chapter 7.

efficiency is given by the solid angle of the objective which is

dΩ4π

=12

(√1− 1

1 + ( 2rd )2

)≈ 1

40. (4.2)

Additional losses of 4% arise due to absorption in the objective and another 6% by a narrowbandpass filter15 used to suppress all light outside the wavelength range 393 nm to 397 nm. In thepresent setup we image the ions on two PMT’s whose outputs are combined with a fast adder andthen counted with a multi-purpose counter card16. In one of the imaging channels a switching box(mirror, 90:10 beamsplitter, coated window) allows us to send all light to the PMT, to distribute itbetween a CCD-camera and the PMT or to send all light to the camera. In front of both PMT’s,at the location of the image, a variable slit aperture17 is installed to suppress stray light. Theimaging plane is about 1.5 m behind the objective which results in a magnification of 24.5. For allexperiments conducted in this thesis, unless mentioned otherwise, state detection was performedwith the PMT.

The quantum state of a single ion is detected with a PMT in the following way. For a given

15PMT II : Semrock FF01-390/18-25; PMT I : Semrock FF01-377/50-23.7-D16National Instruments, PCI 671117Owis, Spalt 40

38

4.5 Fluorescence detection

amount of time, typically 3 ms, the fluorescence lasers at 379 nm and 866 nm are switched onand all photons detected during that time are added up by a counter card18. If the number ofdetected photons is below a certain threshold we assign the D5/2 state to the result, if it is abovethe threshold we assign the S1/2 state. The number of events above/below the threshold relativeto the total amount of repetitions determines the probabilities p1 and p0 corresponding to |α|2and |β|2 of equation (2.1). This state discrimination also works for multiple ions by setting kthresholds and determining the pk probabilities of k fluorescing ions. A typical histogram for thecounts acquired for two ions in 3 ms with 3000 repetitions for an equal superposition of S1/2 andD5/2 can be seen in figure 4.5(a).

For the experiments conducted in chapter 7 it was important that the ions are not heated duringdetection as additional entangling operations had to be carried out after the state detection. Upto three state detections were required during one experiment. Thus a short detection time wasessential in order to prevent a loss of phase coherence of the unmeasured quantum bits due todecohering processes occurring during the detection interval. Finally the detection fidelity hadto stay above 99%. In order to fulfill all requirements we installed a second PMT and added itsoutput to the first one and switched off/shielded all light sources in the lab. With this effort wewere able to detect on average 7.8 photons within 250 µs at a dark count rate of 0.07 when usinga 397 nm laser that had its power and detuning set as for Doppler cooling. This ensured thatthe ions stayed at the Doppler limit during the detection. The signal to noise ratio >100 enabledus to get low conditional probabilities for wrong quantum state assignments. The probability ofdetecting 0 or 1 photons even though the ion was in the bright state, was p(D|S) =0.24%. Theprobability of detecting more than one photon was p(S|D) =0.39% if the ion was in the dark state.

The second device used in some experiments for detecting the quantum state of the ions is anElectron Multiplying Charge Coupled Device (EMCCD)19 camera. The focal plane of the imagingsystem is chosen such that the ions are imaged onto the camera chip. For state detection it ispossible to define a region of interest (ROI) around the positions of the ions of typically 10 × 40pixels for two ions. Only this small region has to be read out, digitized and processed insteadof the full 1024 × 1024 pixels. The spatial information in the fluorescence signal obtained withthe camera enables us to determine not only the number of fluorescing ions but also which ion(s)is(are) fluorescing. For a proper state discrimination two reference pictures are taken: One withall ions fluorescing, the other one with switched-off 866 nm laser for a background picture. Thesetwo pictures are subtracted from each other and then all vertical pixels are added up resultingin a fluorescence image of the ion string that is integrated in the direction perpendicular to thestring. Then a Gaussian is fitted to every single peak. From these fits the 2N different possiblestates are being built serving as reference pictures to be compared with the picture of an ionstring whose quantum states are to be measured. To determine the outcome of an experiment allpictures are treated in a similar way as the reference pictures. First the reference background issubtracted, then all vertical pixels are added up and the result is multiplied with all 2N referencestates. A maximum likelihood method then determines the outcome of the measurement. Withthis method single ion state detection can be done within 5 ms with a fidelity exceeding 99%.State discrimination with the camera is technically more demanding than the PMT detection as18National Instruments, PCI 671119Andor iXon DV885JCs-VP, pixel size 8 × 8 µm, quantum efficiency at 397nm 37%

39

Experimental setup

Figure 4.6.: (a) 397 nm laser setup providing the light for Doppler cooling, state detectionand optical pumping. The EOM at 3.2 GHz generates sidebands on the light to address bothof the 43Ca+ hyperfine ground states.(b) Laser setup for generating the light at 854 nm and866 nm for repumping the D-states. Both light sources are combined on a 50:50 beamsplitterand sent with two polarization maintaining fibers to the vacuum vessel. For efficientlyrepumping the D3/2 state in 43Ca+ two additional AOM’s modulate frequencies at 145 MHzand 245 MHz onto the 866 nm beam.

an additional program is required for controlling the camera and evaluating the pictures taken.Furthermore this program has to be synchronized with the experiment control software to ensureproper state detection.

4.6. Laser system

An advantage of calcium ions is that all necessary wavelength can be produced by solid statelasers. For all dipole transitions commercial diode laser systems from Toptica were employed inthe experiments. The light for driving the S → P transitions was generated by second harmonicgeneration of near infrared diode lasers. All diode lasers are stabilized to medium finesse (F=300)cavities (see Jan Benhelms thesis [94]) with the Pound-Drever-Hall method. The error signal isfed back onto the piezo of the diode laser grating as well as on the current driving the diode. Thisreduces the laser line width to about 100 kHz limited by acoustic vibrations of the cavity mirrors.The frequency of the diode lasers can be tuned by 3 GHz by changing the cavity length with apiezo attached to one of the mirrors. The free-spectral range (FSR) of the cavities is 1.5 GHz. Theonly laser that is not a diode laser is the narrowband titanium-sapphire (Ti:Sa) laser at 729 nm. Itis stabilized to a high finesse cavity in order to achieve a line width below 10 Hz. The experimentalsetup of all laser sources will be described in the following sections.

40

4.6 Laser system

4.6.1. Lasers for Doppler cooling, state detection, optical pumping and

repumping

Laser at 397 nm

For Doppler cooling, state detection, optical pumping and repumping the ions are excited on the397 nm dipole transition from S1/2 to D5/2 (see figure 2.3 and figure 2.4). This light is generated byfrequency doubling a diode laser system20 running at 794 nm. At the output of the doubling stagewe have between 5 and 10 mW of blue light which is split by a polarizing beamsplitter (PBS) intoa σ beam and a π beam (see figure 4.6(a)). Both beams are switched on and off by acousto-opticalmodulators (AOM)21 driven at 220 MHz. In the σ beam line there is an additional electro opticmodulator (EOM)22 creating sidebands on the light. This is necessary to couple both hyperfinemanifolds in the S1/2 ground state of 43Ca+ to the P1/2 hyperfine states. Both light beams aresent with polarization maintaining single-mode fibers to the vacuum vessel. The σ beam is sentonto the ions from the NW viewport with a σ+ polarization while the π beam enters the vacuumvessel either on the SE-viewport or the bottom viewport. An additional frequency-doubled lasersystem is available which can be tuned from 393 nm to 397 nm. Currently it is used to generatea 400 MHz red-detuned beam at 397 nm for Doppler cooling. This beam is needed to efficientlyrecrystalize two ions after a melting of the ion crystal e.g. due to a background collision. It issuperimposed with the σ beam on a PBS having a 90 rotated polarization.

Repumping lasers

For repumping the ion from the D3/2 and P1/2 states light at 854 nm and 866 nm is needed. Thislight is provided by two diode laser systems23. The 854 nm laser is switched on and off by an80 MHz AOM in double pass configuration to ensure proper switch-off(see figure 4.6(b)) of thelaser beam. The 866 nm laser is switched by a single pass 80 MHz AOM. AOM’s at 145 MHzand 245 MHz create additional light beams that are recombined at beamsplitters for efficientlyrepumping 43Ca+. This increases the Doppler cooling efficiency and the fluorescence signal. TheseAOM’s are only used when running the experiment with 43Ca+ ions. The light beams of bothlasers are superimposed on a 50:50 beamsplitter and then sent to the SE and the bottom viewportof the vacuum vessel by two polarization maintaining single-mode fibers.

4.6.2. Ultra-stable titanium-sapphire laser at 729 nm

With a laser at a wavelength of 729 nm we have the capability of exciting the ions on the S1/2 toD5/2 quadrupole transition. Such a laser offers multiple applications, like:

• spectroscopy on the quadrupole transition

• sideband cooling to the motional ground state

• frequency resolved optical pumping20Toptica DL-SHG21Crystal Technology, 3220-12022New Focus, 443123Toptica, DL100

41

Experimental setup

Figure 4.7.: Laser setup to provide 3 different 729 nm light beams and the light used forstabilizing the laser onto a high finesse cavity. The light intensity of the Ti:Sa is stabilizedby feeding back the signal of PD1 to the RF power driving AO1. AO2 is used for intensity,frequency and phase manipulation of the light sent to the ions. AO3 to 5 are used as switchesto guide the light through fibers to different ports of the vacuum chamber. The feedbackto the Ti:Sa using a Pound Drever Hall scheme is threefold. A low frequency feedback ontothe tweeter and the Brewster plate as well as a mid and a high frequency feedback to anintracavity EOM is applied.

• coherent manipulation of optical qubits

• optical shelving for state detection

• state transfer and initialization

These tasks are very demanding regarding the laser’s frequency and power stability. Furthermoreit necessary to tune the laser over 100 MHz within a microsecond and set it to different transitionsfor 43Ca+ and 40Ca+ which have a frequency difference of 5.5 GHz. The setup of the laser and itsperformance is described in [94] and [118]. Here the idea of the locking scheme is recapitulatedand the optical setup is described.

To meet all requirements listed above, the laser is stabilized to a vertically mounted [119] highfinesse cavity24(F = 412 000, line width = 4.7 kHz, FSR = 2 GHz) made out of ultra low expansionmaterial (ULE). It rests on teflon feet inside a temperature stabilized (± 1 mK) vacuum tank(10−8 mbar). The vacuum tank is inside a temperature stabilized wooden box shielding the cavityfrom acoustical perturbations. Typical drift rates of the cavity measured by referencing it to theions (see chapter 5) are 3 Hz/s or less which means that the cavity spacer length changes withina year by no more than 20 nm.

This high stability comes at the price of being unable to tune the cavity as both mirrors areoptically contacted to the ULE spacer. In order to get the necessary tuning range we use an AOM25

with a center frequency of 1.5 GHz and a tuning range of 1 GHz in double-pass configuration. With24Advanced Thin Films, CO, USA25Brimrose GPF-1500-1000

42

4.6 Laser system

this AOM, located between the laser output and the high finesse cavity (see figure 4.7 AO6), weare thus able to tune the laser to any desired frequency. The AOM also allows us to cancel thedrift of the cavity by comparing the laser frequency with the ions transition frequency and feedingback on the frequency of the AOM.

The optical setup and the servo loop for the Ti:Sa laser are shown in figure 4.7. The feedbacksignal is generated by a Pound-Drever-Hall (PDH) [120] locking scheme. An EOM26 modulatessidebands onto the light and the light reflected by the cavity creates the error signal detected bya photodiode (P4). The locking scheme of the laser consists of three parts. Slow fluctuationsare actively canceled by sending the error signal through a PI-servo and applying it to the piezo(tweeter) moving one of the Ti:Sa cavity mirrors and the scanning brewster plate. This loophas a servo bandwidth of 10 kHz limited by the mechanical resonances of the piezo. A higherservo loop bandwidth is achieved by inserting an EOM inside the laser cavity. By changing therefractive index of the EOM the effective cavity length is stabilized. On one electrode of the EOMwe apply the error signal sent through a fast proportional amplifier27(10 MHz bandwidth). Theother electrode is connected to a high voltage amplifier28 (up to 400 V) which receives as an inputthe error signal sent through another PI-servo. The grounds of the two amplifiers are connected,thus the EOM is floating. With this setup we get servo bandwidths of 300 kHz for the high voltagepart and 1.6 MHz for the fast amplifier.

The laser and the high finesse cavity are coupled by a polarization maintaining single-mode fiber.To avoid a frequency broadening of the light transmitted through the fiber by acoustic noise, weimplemented an active fiber noise cancelation [118, 121]. The light passing the fiber is first sentthrough a 50:50 beam splitter. One part is retro-reflected by a mirror and sent onto a photodiode.The second part is sent through an 80 MHz AOM (see figure 4.7 AO7) and then through the fiber.The end-facet of the fiber is at right angle, thus 4% of the light intensity are back-reflected. Theback-reflected light again passes the AOM and hits the same photodiode as the incoming beam.The beat note at 160 MHz contains the phase fluctuation imprinted on the light by the fiber. Anerror signal is created by comparing the beat note to a stable frequency source29. By implementinga phase locked loop, the frequency of the beat signal is kept in phase by feeding back onto a voltagecontrolled oscillator providing the input signal of the AOM. The light power sent through the fiberis power-stabilized by monitoring the power transmitted through the high finesse cavity. In thisway beam pointing and polarization variations are also canceled such that the PDH error signal isnot affected.

For the experiments described in chapters 6 to 8 the line width of the 729 nm laser and its spectralpurity plays a crucial role. To determine these quantities we performed a beat measurement witha similar laser located at another institute 500 m away. A fiber was used to transmit the lightfrom one building to the other. To avoid any line width broadening of the light we employed afiber noise cancelation. The slow relative drift of the lasers was canceled by applying a continuousslow frequency change to AO7. The spectrum of the beat signal at a center frequency of 10.8 MHzwas recorded with a spectrum analyzer. Figure 4.8(a) shows the power spectrum of the beat for

26Linos/Gsanger PM2527Femto HVA-10M-60-F28base on Apex, PA9829Rhode & Schwarz, SML01

43

Experimental setup

Figure 4.8.: (a) Beat measurement of two 729 nm laser connected via a 500 m long noisestabilized fiber. A Lorentzian fit yields a FWHM line width of 1.8 Hz within a measurementinterval of 4 s. The inset shows the result of Ramsey experiments on the magnetic fieldinsensitive transition S1/2, F = 4,mF = 4 → D5/2, F = 4,mF = 3. of 43Ca+. The contrastof the Ramsey experiments is plotted versus the Ramsey waiting time. The red line is aGaussian fit to the date with a FWHM of 8.1 ms reflecting the loss of coherence due to thefinite laser line width and additional frequency noise introduced by a 2 m fiber guiding thelight to the ions.(b) Power spectral density of the laser beat with a resolution bandwidth of1 kHz where the reference laser was spectrally cleaned with a filter cavity. The transmissioncurve of the filter cavity is given by the blue dashed line. The measurement is limited bythe dynamic range of the spectrum analyzer and the reference laser as can be seen from thenormalized spectrum of the error signal. An integration of the power spectrum shows thata fraction of less than 10−4 of laser light power is outside a ±250 Hz window around thelaser’s carrier frequency.

a 4 s acquisition time and a resolution bandwidth of 1 Hz. A Lorentzian fit yields a FWHM linewidth of 1.8 Hz. If we assume both lasers have the same Lorentzian frequency spectrum we get aline width for each laser of 0.9 Hz.

To get a better picture of the spectral features of our laser, the spectrum of the second laser wascleaned with a filter cavity. The transmission function of this cavity is given in figure 4.8(b) togetherwith the normalized power spectral density of the beat measurement for a resolution bandwidthof 1 kHz. The figure also shows the spectrum of the laser lock error signal. By comparing bothspectra we can easily identify the characteristic humps in the spectrum as “servo bumps“ of thelaser. Furthermore we can conclude that the beat measurement is limited by the reference laserand the dynamical range of the spectrum analyzer. An integration of the power spectrum showsthat a fraction of less than 10−4 of laser light power is outside a ±250 Hz window around thelaser’s carrier frequency.

For time scales larger than the 4 s used to determine the spectrum the beat is dominated byan oscillatory frequency variation with a few seconds period. This accumulates to an effectiveline width of 50 Hz in a 4 h acquisition time. This is also consistent with Ramsey measurementsperformed on the magnetic field insensitive transition S1/2, F = 4,mF = 4 → D5/2, F = 4,mF = 3.

of 43Ca+ (see figure 4.8(a) inset). From the Gaussian fit to the data we determine a FWHM decaytime τ1/2 of the Ramsey contrast of 8.1 ms. This decay time can be directly transferred into an

44

4.7 Computer control and RF generation

Figure 4.9.: Sketch showing the interconnection of different hardware components. Theexperiment is controlled by a LabView programm running on a windows computer (mainPC). Additionally a computer controlling the camera and performing automatized dataevaluation is connect via a TCP/IP connection. The main computer controls the remaininghardware, reads out the counter data of the PMT and programs the versatile frequencysource (VFS). The VFS is triggered by the AC power line and provides TTL signals and allphase-synchronous radio frequency signals.

effective laser line width with [122]

∆ν =2 ln 2

π

1τ1/2

. (4.3)

From this formula we can infer a laser line width on the ions of 55 Hz for a measurement time of10 Minutes.

The output of the laser is intensity stabilized by monitoring a small portion of the light bya photodiode30 (PD1), comparing it with a stable voltage reference and feeding back the errorsignal onto the RF power driving AO1. The frequency and intensity of the light sent to the ions iscontrolled by AO2. We can choose between three different fiber ports which are switched on andoff by AO3 to 5. AO3 can be driven simultaneously by two frequencies to create a bichromaticlight field. Typically the frequency difference between the two beams is 2.4 MHz which leads to adiffraction into slightly different directions with an angular separation as small as 0.025 such thatthe coupling efficiency to the single mode fiber is reduced by about 15% compared with a singlefrequency beam.

4.7. Computer control and RF generation

Qubits in ion trap quantum information experiments are manipulated by laser pulses whose powerand length have to be controlled. Furthermore phase coherence has to be ensured for all pulses.This can be achieved by using AOMs which directly transfer the frequency, power and phaseinformation from the radio frequency onto the light. In this way, the problem is reduced togenerating precisely timed digital signals and phase-coherent radio frequency pulses with adjustablelength and amplitude. For this purpose a versatile frequency source [105, 123] (VFS) was developed.It is based on a field programmable gate array (FPGA) which controls a direct digital synthesis(DDS) board. This DDS board can generate pulses with 16 different frequencies in the range of

30Thorlabs PDA 100A-EC

45

Experimental setup

0-300 MHz and allows for phase coherent switching between them. The pulses have a minimalduration of 10 ns but are typically in the range of 1 µs to 1 ms. The amplitude and shape ofthe pulses are controlled by a variable gain amplifier. The frequency range of the DDS board islimited by filters to 300 MHz. Higher frequencies like the 3.2 GHz for driving the hyperfine qubitare created by appropriate mixing and filtering of the signals. The frequency resolution is about0.1 Hz. Additionally up to 16 logic channels (TTL) can be used which are also set by the FPGAcontroller. These TTLs are used to control all fast and precisely timed switching required duringan experimental cycle. Logic signals can be sent to the FPGA via 8 input channels.

The FPGA is connected to a PC (main PC) by an ethernet connection and is programmed via apython31 server. Figure 4.9 shows a sketch of the interconnections between the different hardwarecomponents. The software running on the main PC controlling the experiment is written inLabView. This program is used to send command sequences to the FPGA board which are executedas soon as it receives a trigger on one of its input channels. Sequences run synchronously with the50 Hz power line cycle to avoid varying magnetic fields from one execution to the next. This isagain insured by a trigger to one of the input channels. The LabView program also controls severalNational instrument cards32. They are used to generate analog signals for controlling the powerand frequency of the dipole lasers. TTL outputs of these cards are used to switch time-noncriticalsignals.

The PMT signals acquired by the counter card during an experiment are read out, processed anddisplayed by the LabView program. The program can be used to run automatized sequences. Thedata acquired during such sequences can be sent to the camera PC for data evaluation and modelfitting. The values determined by these procedures are then fed back into the LabView program.A typical application consists in calibrating the duration of all pulses required in a sequence.

Another LabView program running on the camera PC controls the camera hardware and isresponsible for read-out and evaluation of the camera pictures. The programs on both computersare communicating via ethernet.

Signal generators are computer controlled by a GPIB-Bus33. The ISEG high voltage sourceproviding the tip voltage is connected by a CAN-bus34 to the control PC.

To ensure phase-coherence between all RF-signals the signal generators as well as the VFS-boxare referenced to a 10 MHz GPS assisted quartz oscillator35.

31High-level programming language32National Instruments PCI 6711, PCI 6703, DIO 6433IEEE-48834Controller area network, serial bus standard35Menlo Systems GPS 6-12

46

5. Experimental techniques

This chapter will outline techniques used on a daily basis in the laboratory. It will explain how toload ions into the trap and detail the structure of a typical experimental sequence. An importantpoint is the referencing of the 729 nm laser to the ions, which is one of the key elements fordoing high precision spectroscopy and QIP experiments. Then implementation of arbitrary qubitoperations will be demonstrated with the help of ion shuttling and the capability of single ionaddressing. How to analyze the state of an ion by state tomography will be presented at the endof this chapter.

5.1. Loading ions by photoionization

Typically, ions are loaded by photoionization of an atomic beam produced by an oven. The ovenconsists of a stainless steel tube which is heated by an electric current. The tube is about 8 cmlong to ensure a proper collimation of the beam through the trap center, thus avoiding excessivedeposition of calcium on the trap electrodes. The photoionization beam is sent through the trapcenter at a slightly different angle. The ions are directly ionized inside the trap volume. Comparedto loading by electron impact it has several advantages. Electron bombardment creates patchcharges on the trap which have to be compensated in order to avoid micromotion. It requires ahigher atomic flux to get a decent loading rate because this ionization process is five orders ofmagnitude smaller as compared to photoionization. This leads to bigger calcium layers on the trapsurfaces which increases the patch charge problem. Furthermore, electron bombardment is notisotope selective and even atoms from the background gas can be ionized. Especially this point isof importance as we want to load 40Ca+ and 43Ca+ deterministically.

Calcium is photoionized by a two step process (see figure 5.1(a)). A laser in Littrow configurationat 423 nm resonantly excites the atom from the 4s1S1 to the 4p1P1 state. From there it can beionized by a second step using light with a wavelength smaller than 390 nm. In the experimentwe use a free-running laser diode at 375 nm. Both beams are superimposed on a beamsplittercube and sent through a single-mode fiber towards the ion trap via the SE-viewport of the vacuumchamber (see figure 4.3). Typically 50 to 100 µW for the 423 nm laser and about 500 µW for the375 nm laser are used for efficient loading.

Isotope selective loading can be achieved by tuning the 423 nm laser to the 4s1S0 ↔ 4p1P1

resonance (see figure 5.1(b)) of the respective isotope. The frequency of the 423 nm laser can bedetermined either on the wavemeter or by saturation spectroscopy on a calcium vapor cell. Thespectroscopy clearly shows the Doppler-free resonance for the most abundant calcium isotope 40Ca.By detuning the laser by 612 MHz to the blue it becomes resonant with the 43Ca resonance. Dueto the finite Lorentzian line shape and the natural abundance of calcium the relative loading rate of

47

Experimental techniques

Figure 5.1.: (a) Two step photoionization of calcium. The 423 nm laser can be used tomake the process isotope selective. (b) Excitation spectrum of calcium at 423 nm. Figure istaken from [124]. The numbers above the curve show the resonances of the respective calciumisotopes. 43Ca+ has an isotope shift of 612 MHz with respect to the 40Ca+ resonance. Thered line is fit to the measured data taking into account a single resonance for 40Ca+. Theresiduals between the red and the black curve are shown in green and indicate the resonancesof other isotopes.

43Ca+ compared to other isotopes is only 50%. To further increase the isotope selectivity there aretwo ovens in the apparatus. One is filled with calcium granules with a natural isotope abundanceof 97% 40Ca and the other is filled with an enriched source consisting of 81% 43Ca, 13% 40Ca and5% 44Ca.

For loading calcium ions the Doppler cooling laser at 397 nm, the repumper at 866 nm and bothphoto-ionization lasers are continuously on. The current in the oven is set such that about twoions are loaded per minute. By monitoring the fluorescence signal with the PMT and/or camera,ions appearing in the trap can be detected. Larger crystals are loaded by waiting for longer times.In case that too many ions are in the trap all of them have to be discarded by switching off theRF drive. Then the loading procedure is started again.

After the intended number of ions has been loaded, the Doppler cooling lasers are adjusted foroptimum detuning and power [92]. Two different power settings for the 397 nm laser have to beoptimized: One for proper Doppler cooling, to get as close as possible to the Doppler-limit andthe other for a maximum fluorescence to get a good state detection. The parameters for the twosettings will be given in the following section. When all dipole lasers are properly set we can startrunning experimental sequences manipulating the qubit.

5.2. Experimental sequence

All experiments are executed in a pulsed fashion. Such a sequence typically consists of six buildingblocks (see figure 5.2) which are described in the following. A single experiment is typically repeated50-200 times.

48

5.2 Experimental sequence

Figure 5.2.: Typical sequence of laser pulses executed in an experimental cycle. (1) Dopplercooling, (2) optical pumping, (3) sideband cooling, (4) frequency resolved optical pumping,(5) quantum state engineering, (6) state detection. Each sequence is triggered by a TTLpulse (0) synchronized with the line cycle and is typically repeated 50, 100 or 200 times.

(1) Doppler cooling

For 40Ca+, it is sufficient for efficient Doppler cooling to use a single 397 nm beam couplingS1/2 → P1/2 and a single 866 nm laser repumping the transition D3/2 → P1/2. To cool all threenormal modes of motion of the ion in the trap, care has to be taken that the Doppler-coolingbeam has sufficient overlap with all modes. The power of the 397 nm laser is set such, that itsRabi frequency is half the saturation intensity [92] of the dipole transition. It is red detuned withrespect to the resonance by half the linewidth of the transition. The 866 nm laser power is set suchthat the fluorescence is just below saturation to avoid additional line broadening. It is slightly bluedetuned to avoid coherent population trapping. The degeneracy of the Zeeman sublevels is liftedby a magnetic field.

For 43Ca+ Doppler cooling is a little bit more complicated. Here three light fields for the397 nm laser are necessary. Two of the light fields are σ+- polarized, where one is couplingthe S1/2, F = 4 → P1/2, F = 4 transition and the other one, detuned by 3.2 GHz, is coupling theS1/2, F = 3 → P1/2, F = 4 transition. The 3.2 GHz component is created by modulating sidebandson the light field with an EOM (see figure 4.6). The third light field is a π polarized beam whichis needed to avoid pumping into the dark state S1/2, F = 4,mF = 4.

In the experiment it turned out that the π and the σ+ beam should not have the same frequency.If they do, Doppler cooling is not working properly which might be due to dark states that areinefficiently cooled. A detuning of 5 MHz to the red for the π beam, with respect to the σ+ lightfield, turned out to yield the best Doppler cooling results. The 866 nm laser is tuned close to theD3/2, F = 3 → P1/2, F = 3 transition. For efficient repumping additional light fields shifted by-145 MHz and -395 MHz are provided (see figure 4.6). In this way all D3/2 hyperfine levels arecoupled to one of the P1/2 states. Power and detuning from the resonances for both lasers are setthe same way as for 40Ca+. A magnetic field lifts again the degeneracy of the Zeeman states.

The fluorescence collected during Doppler cooling is used to check if the ions are in a crystal-likestructure. If the fluorescence collected for a single sequence falls below a certain threshold thewhole experiment is discarded and executed again.

49


(2) Optical pumping

A weak short pulse with a σ+ polarized 397 nm laser together with an 866 nm laser pulse is usedfor optically pumping the ions into either S1/2,mj = 1/2 for 40Ca+ or S1/2, F = 4,mF = 4 for43Ca+. As for Doppler cooling, more light fields have to be used for pumping 43Ca+. After thisstep we achieve a pumping fidelity of 99% into the desired state within 70 µs for 40Ca+ and about98% for 43Ca+ ions. The pumping fidelities are limited by beam polarization and overlap of the kvector of the light field with the magnetic field axis.

(3) Sideband cooling

Sideband cooling to the motional ground state is done in the same way for both isotopes. The729 nm laser is tuned to the red sideband of the stretched state S1/2,mj = 1/2 → D5/2,mj = 5/2for 40Ca+ and S1/2, F = 4, mF = 4 → D5/2, F = 6,mF = 6 for 43Ca+. Every state transfer fromS1/2 to D5/2 reduces the mean phonon number by one quantum. To reduce the lifetime of theD5/2 level a 854 nm laser is switched on at the same time. The population is transferred from theD5/2 state to the P3/2 state from where it decays back to the ground state. To avoid populationleaking from the cooling cycle due to a decay into D3/2, an 866 nm laser is continuously on. Thisrepumping via the P1/2 state does not guarantee a decay into the desired S1/2 state. Additionaloptical pumping with the 397 nm laser brings the population back into the cooling cycle. Afterabout 7 ms a mean phonon number of 〈n〉 = 0.05(5) is reached.

(4) Frequency resolved optical pumping

To further increase the optical pumping efficiency we make use of the 729 nm laser. The schemefor 40Ca+ is similar to sideband cooling. For a duration of 500 µs the 729 nm laser is switched ontogether with the 866 nm and 854 nm laser. The quadrupole laser couples the states S1/2,mj =−1/2 → D5/2,mj = 3/2 depleting the undesired Zeeman state. Due to the repumping the popu-lation piles up in the S1/2,mj = 1/2 and we get pumping efficiencies exceeding 99.8% for a singleion.

In 43Ca+ we transfer the population in S1/2, F = 4, mF = 4 to the D5/2, F = 6, mF = 6state. Then we again apply an optical pumping step with the 397 nm laser. This redistributesthe population remaining in the ground state. Another π pulse exchanges the populations inS1/2, F = 4,mF = 4 and D5/2, F = 6, mF = 6. After this step 98% of the population shouldbe in S1/2, F = 4,mF = 4 and the rest in D5/2, F = 6, mF = 6. A light pulse with 854 nmlaser clears out the D5/2 state via the P3/2, F = 5,mF = 5 from where it can only decay intoS1/2, F = 4,mF = 4. With this procedure we achieve pumping fidelities for a single 43Ca+ ion ofmore than 99.4%.

(5) Quantum state manipulation

After the ions are prepared in the motional ground state and the desired electronic state, the qubitin 40Ca+ and 43Ca+ can be manipulated by 729 nm laser pulses. If the qubit is encoded in thehyperfine states of 43Ca+ it can also be manipulated by microwave pulses.

50

5.3 Referencing the 729 nm laser to the ions

Figure 5.3.: Variation of the magnetic field over the line cycle. The data points weremeasured by delaying the start of a Ramsey experiment with respect to the line trigger.The strong 50 Hz component is most likely caused by transformers and other electronics ofthe setup.

(6) State discrimination

By detecting the fluorescence of the ion on either the camera or the PMT one can distinguishbetween the S1/2 level and the D5/2 level. The power of the 397 nm laser is increased for a highfluorescence rate to get a good state discrimination within a few milliseconds. To discriminatebetween the states of the hyperfine qubit S1/2, F = 4,mF = 0 and S1/2, F = 3,mF = 0, additionaltransfer pulses are used to hide the S1/2, F = 4,mF = 0 population in the D5/2 manifold. Twotransfer pulses to different D5/2 Zeeman states ensure that the detection efficiency is above 99%.

5.3. Referencing the 729 nm laser to the ions

In order to make use of the 729 nm laser for QIP experiments an exact knowledge of the magneticfield and the laser frequency is necessary. Therefore the laser is referenced to the ions in the trap.The referencing is done by probing two different transitions (a,b) in either 40Ca+ or 43Ca+. Fromthe exact knowledge of the Zeeman splitting of the transitions one can then infer the relative laserfrequency and the magnetic field. By recording the transitions for several minutes it is possibleto determine the drift of the reference cavity and, by active feed forward, to stabilize the outputfrequency of the laser up to some small fluctuations caused by magnetic field variations.

The ions are Doppler-cooled and initialized by optical pumping. Then a Ramsey type scheme [125]is employed to determine the transition frequency. A π/2 pulse creates a superposition of the twolevels to be probed. Then after a Ramsey waiting time τR, a second π/2 pulse is applied. One tran-sition is probed twice in this way. The first time the second pulse has a relative phase of φ1 = π/2and the second time it has a phase of φ2 = 3π/2. After each measurement the population of thelevels is measured as described in chapter 4.5. By repeating this sequence a hundred times one caninfer the probabilities of finding the ion in the excited state pφ1 and pφ2 for the respective phases.From this one can determine the frequency difference between the laser and the probed transitionas [94]

∆ν/(2π) =1

2π (τR + 2τπ/π)arcsin

pφ1 − pφ2

pφ1 + pφ2

. (5.1)

Where τπ denotes the pulse length to drive a π-pulse on this transition with the chosen laser

51


power. Typical values for τπ are between 5 and 20 µs whereas τR is typically 10 times longer, thatis 0.1-1 ms.

From the deviation of the transition frequencies ∆νa and ∆νb we first determine the magneticfield. From this knowledge and the frequency of one transition we then determine the transitionfrequency at zero magnetic field. This serves as a reference from where all other transitionsfrequencies can be calculated for a given magnetic field.

The referencing measurements and the analysis are fully automatized and run about everyminute. The measurements are recorded over time and a polynomial fit to the frequency datareveals the drift of the cavity. Usually a first order fit is employed and an extrapolation is used tofeedback on the signal generator producing the frequency for the AOM set up between laser andcavity (see figure 4.7 AO6). With this feedback scheme the laser frequency deviates by less than200 Hz (rms-deviation) from the transition frequency. The exact value depends on τR which islimited by the coherence time of the ions, the linearity of the cavity drift and the line width of theprobe laser.

The magnetic field data are averaged over time where data older than 5 min are not taken intoaccount. The average is used as a a best guess for the magnitude of the magnetic-field during thenext minute. The limiting factor in determining the exact magnetic field is given by erratic jumpsin the magnetic-field by 100-250 µG due to external magnetic fields (elevator etc.). Without thesejumps the magnetic field is stable and can be predicted within about 50 µG over several hours ifthe experiments are triggered to the line cycle.

If the time between the start of an experiment and the line trigger is varied, changes of themagnetic-field by about 1.4 mG are visible (see figure 5.3). To avoid this variation from shotto shot, experiments are always started at a fixed phase of the power line cycle. Nevertheless,since some of the transitions used exhibit a magnetic field sensitivity of 2.8 MHz/G frequencycorrections of up to 4.5 kHz have to be taken into account over the course of a sequence that lastsseveral milliseconds. To avoid these corrections for short sequences the experiments are triggeredto the maximum in the line cycle (around 250) such that the magnetic field does not change infirst order. Ways to get rid of this limitation are either technical, by active or passive magneticfield stabilization, or by choosing a qubit which is insensitive to magnetic field changes [102, 126].Such a magnetic field insensitive qubit was used in chapter 6 to store entangled states for tens ofmilliseconds.

5.4. Implementation of arbitrary qubit operations and ion

shuttling

There are different approaches to carry out single-qubit rotations. A requirement that is alwayspresent is the capability to limit interactions to individual qubits. In the ion trap case this isequivalent to shining in the laser on one ion at a time. One possibility to achieve this is to usemicro-fabricated segmented traps to manipulate the ion motion [127]. With the help of differenttrap electrodes a large ion string can be split into smaller ones and brought to trap zones wherethe selected ions interact with laser fields [127]. This approach requires a high degree of controland stability on the voltages applied to the trap electrodes. Some experiments have demonstrated

52

5.4 Implementation of arbitrary qubit operations and ion shuttling

Figure 5.4.: (a) Laser beam setup for the 729 nm laser. The tightly focused beam 2 is usedfor addressed σz rotations. The beams 1 and 3 are for simultaneous operations on up to threeions. Theses beams are either bichromatic light fields for implementing a MS-interaction orsingle frequency for carrier rotations on all ions. The NE beam is σ+ polarized for driving∆m = ±1 transitions with maximal efficiency. The NW beam is linearly polarized fordriving ∆m = ±2 transitions with maximal efficiency. (b) Pulse scheme to perform a π/2carrier rotation on ion 1 and no rotation on ion 2. Both ions are first rotated by π/4, thenion 1 gets a π phase shift with the AC-stark pulse. The second π/4 pulse on both ionsrotates ion 2 to the initial state and completes the π/2 rotation on ion 1. (c) Pulse schemeto perform a MS-gate on two out of three ions. The first π/4 MS-gate partially entanglesall three ions. A stark shift pulse on ion 1 changes the phase of this ion by π. The final π/4MS-gate entangles ions 2 and 3 and reverts ion 1 to its initial state.

the ability to deterministically split [128, 129] and reorder an ion string [130]. To do this splittingfast the traps have to be small as the trap frequencies scale as d−9/10 with d the distance of theions to the trap electrodes [131]. A problem all traps face is that the heating rate scales as thed−4 [132, 133] which is especially bad for micro-fabricated traps. Although experiments showedthat cooling the traps [134] reduces the anomalous heating a considerable technical overhead isrequired to do QIP with these traps.

Another way to single out ions from a larger crystal is to use strongly focused light beams.Ideally the focus should be much smaller than the separation of two ions which is typically a fewµm. However, even for perfect optics the Gaussian beam profile will lead to some light on theneighboring ion and thus some residual coupling Ωres. With our addressing optics for the 729 nmbeam (see figure 4.4) we are able to get the ratio ε = Ωres/Ω for carrier pulses down to about 5%.This ratio can be further improved by using Stark shift pulses. As the interaction strength forthese scales with the intensity instead of the field strength the addressing error scales now withε2. Addressing errors are thus reduced to below 0.3%. One pays for this increased addressingcapability by an increased sensitivity to intensity fluctuations.

Single-qubit light-shift gates U(i)z (θ) = e−i θ

2 σ(i)z are realized by an off-resonant strongly focused

beam impinging (see beam 2 in figure 5.4(a)) on ion i at a 90 angle relative to the trap axis. Asthe k-vector of this beam is perpendicular to the trap axis this configuration does not allow tocouple to the axial center of mass motion of the ions and can thus not be used to do two-qubitgate operations. A second beam at a 45 angle relative to the trap axis (see beam 1 or beam 3in figure 5.4(a)), illuminating all ions with equal strength serves to carry out gate operations thatare symmetric under qubit exchange. Collective single-qubit gates U(θ, φ) = e−i θ

2 (σ(1)φ +σ

(2)φ ), are

53


Figure 5.5.: (a)Pulse sequence describing a Ramsey experiment on the motional statesof the ion. (b) Ramsey contrast of the experiment versus number of events shuttling theion by 10 µm. The inset shows the contrast of a Ramsey phase experiment on the motionwithout (•) and with (N) a single ion shuttle.

realized by resonantly exciting the qubit transition and controlling the phase φ of the laser light.If instead a bichromatic light field is used for implementing a Mølmer-Sørensen interaction, thepropagator UMS(θ, φ) = e−i θ

2 σ(1)φ ⊗σ

(2)φ is realized. This interaction is used for entangling ions.

As the tightly focused beam is static we still have to move the ion string to address differentions. The shuttling of the ions is done in such a way, that the population of the harmonic oscillatormode does not change during the transport. The curvature of the trapping potential is keptconstant during the movement of the ions. The movement of the ions is achieved by changingthe tip voltage with the circuit presented in chapter 4.1. To avoid energy transfer the ions areaccelerated adiabatically with respect to the trapping frequencies. Typical shuttling times for10 µm of 40 µs corresponding to about 50 axial oscillations were achieved without affecting thepopulation of the COM mode. This was confirmed by performing a Ramsey experiment on the twolowest motional states in the following way: A single ions was cooled to the motional ground stateby sideband cooling. A π/2 pulse on the carrier followed by a π pulse on the red sideband createsthe superposition state |↓, 0〉+ |↓, 1〉. Then the ions were shuttled over a distance of 10 µm alongthe trap axis. After a waiting time of 40 µs the ions were shuttled back to the original position.Again a waiting time of 40 µs ensured that the voltage on the electrodes was settled. The sequenceapplied to the ion is shown in figure 5.5(a). The first two pulses were then applied in reverse orderand the phase of the carrier pulse was scanned to record Ramsey fringes (see figure 5.5(b) inset).The contrast of these fringes was then compared to the case where the ion was not transported. Ifthe ion was heated during the transport, i.e. the occupation of the motional mode was incoherentlychanged, the contrast of the Ramsey fringes decreases with respect to the stationary case. No sucheffect could be observed even for moving the ion 32 times (see figure 5.5(b)).

It can be shown [135] that combining this set of elementary gates U (i)z (θ), U(θ, φ), UMS(θ, φ)

arbitrary multi-qubit operations and mapping of observables Aij to σ(k)z for read-out can be im-

plemented. This is similar to NMR techniques [15] where the spin-spin interaction that is alwayspresent can be switched off by refocusing techniques. The dynamics of the system is thus controlledby single qubit operations. In the ion trap case, the spin-spin interaction is replaced by the MS-gate and the refocusing pulses are implemented with the addressed beam. Two simple examplesfor such a pulse sequence with ions and their effective circuit diagram are shown in figure 5.4 b

54

5.5 State tomography

Figure 5.6.: Absolute values of the density matrices for the states (a) |↑↓〉+|↓↑〉and (b)|↓↓〉+|↑↓〉reconstructed by a maximum likelihood method.

and c.The sequence in figure 5.4 b shows how to do a π

2 pulse on only one of two ions. Both ions arefirst rotated by U(π

4 , 0), then ion 1 gets a phase shift with the AC-stark pulse U(1)z (π). The second

U(π4 , π) pulse on both ions rotates ion 2 to the initial state and completes the π/2 rotation on ion

1. Example c is similar to the NMR case and is used to entangle two out of three ions. The firstUMS(π

4 , 0) MS-gate partially entangles all three ions. A stark shift pulse on ion 1 U(1)z (π) changes

the phase of this ion by π. The final UMS(π4 , 0) MS-gate entangles ions 2 and 3 and reverts ion 1

to its initial state. So by flipping the phase of one of the ions the interaction is switched off.

5.5. State tomography

State tomography is used to determine the density matrix of a quantum state by measurements onan ensemble of identically prepared quantum states. As described in chapter 2.1.5, a measurementof the expectation values λi of the Pauli operators σx, σy, σz is sufficient to directly reconstruct adensity matrix describing a qubit state. For a qubit encoded in the electronic state of a trappedion, σz can be directly measured by detecting the fluorescence of the ion (see chapter 4.5). Theother two observables are measured by applying a unitary transformation prior to measuring σz onthe transformed state. This transformation maps the eigenvectors corresponding to the eigenvaluesof σx or σy onto the eigenvalues of σz (see chapter 2.1.3).

For multi-qubit systems it is necessary to measure observables which are tensor products ofPauli matrices. The measurement procedure requires single-ion addressing and single-ion detectioncapabilities. For two ions, observables like σ1

z⊗σ2y are determined by applying an addressed rotation

on qubit 2 prior to measuring σ1z , σ2

z (each (±1)) with the CCD-camera. By correlating the resultsthe expectation value for σ1

z ⊗ σ2y can be measured by multiplying the ±1 outcomes and averaging

over all realizations. To reconstruct the density matrix of two qubits nine measurements have to becarried out to determine the 16 expectation values of all products of σi⊗σi with σi = I , σx, σy, σz.

The unitary transformations we apply to the ions are implemented as shown in the precedingchapter. A unitary transformation to measure e.g. σ1

z ⊗ σ2y would exactly look like the sequence

of pulses shown in figure 5.4(b)Ux

(π

4

)U1

z (π)Ux

(π

4

). (5.2)

55


A problem one faces is that in an experiment one can never determine the observables exactly,as an infinite number of measurements is necessary to eliminate the statistical error. Thus ameasurement of an expectation value λi is only a best estimate for the actual expectation value.This seemingly small difference has important consequences as real physical Bloch vectors haveto lie inside the Bloch sphere. When a state is reconstructed that is close to the boundary thestatistical fluctuations might lead to a single-ion Bloch vector lying outside the unitsphere. Inthis case the corresponding density matrix has one eigenvalue > 1 and one eigenvalue < 1. Asthis matrix is no longer positive semidefinite it does not describe a physical state. This problemgets more severe the higher the dimension of the Hilbert-space becomes as more and more densitymatrix entries will be close to zero and can easily get negative if not enough data are taken.

This problem can be overcome by making use of the measurement results in another way. Thestrategy we use to determine the density matrix is based on a maximum likelihood method [136].This method searches among all physically possible density matrices the one that is most likelyto reproduce the observed measurement results. The measurements described above project thequantum state ρex onto a set of different bases. For each of these bases, N copies of ρex were used,and in Nfi experiments ρex was projected onto the state |Ψi〉. The probability of finding exactlythese results for an arbitrary density matrix ρ is given by the likelihood function

L(ρ) =∏

j

〈Ψj | ρ |Ψj〉Nfi . (5.3)

The maximum likelihood estimation for an experimentally obtained density matrix is given bythe state ρ that maximizes the log-likelihood function

L(ρ) = N∑

j

fj log 〈Ψi| ρ |Ψi〉 . (5.4)

As L(ρ) is a convex function on a convex set, it has no local maxima which simplifies the taskof maximizing L(ρ). If a directly reconstructed density matrix is valid it will also maximize thelikelihood function, thus both methods reconstruct the same density matrix. Different numericalmethods can be used to maximize the log-likelihood function. The method used to reconstruct alldensity matrixes in this thesis relies on an iterative procedure [137] solving a nonlinear operatorequation.

Figure 5.6 shows the absolute values of the density matrices for the states |↑↓〉+|↓↑〉 and|↓↓〉+|↑↓〉 created in the experiment.

56

6. Entangling calcium ions

The most complicated operations needed for QIP in ion traps are universal multi-qubit operations.Most implementations so far lack the performance to carry out gate operations in series withoutproducing too high errors. This chapter will describe experiments on entangling either a pair of40Ca+ ions or a pair of 43Ca+ ions. The first section will focus on a high-fidelity entanglementof 40Ca+ ions cooled to the ground state and to thermal states of motion with 〈n〉 6= 0. Theentanglement of three 40Ca+ ions by means of a MS interaction and the implementation of thefirst steps to do arbitrary quantum circuits on three ions will be discussed in the next section. Thechapter will be concluded by the description of an experiment where the knowledge gained with40Ca+ ions was transferred to the more complicated level structure of 43Ca+ and used to entanglea pair of 43Ca+ ions. Especially interesting is the mapping of the entanglement from the opticalonto the hyperfine qubit.

6.1. High fidelity two ion Bell states

Recently the first application of a Mølmer-Sørensen gate operation to an optical qubit was demon-strated [40]. In this experiment Bell states with a so far unmatched fidelity of 0.993(1) weredeterministically created. Here, a further investigation of this universal gate operation acting onoptical qubits with an extended experimental analysis is presented. Particular emphasis is put onthe compensation of AC-Stark shifts and amplitude pulse shaping to reach high fidelities withoutcompromising the gate speed substantially. The gate characterization is extended further by inves-tigating the fidelity decay for different input states after up to 21 individual operations. Possibleerror sources for the gate operation will be listed at the end.

Moreover, the first experiments demonstrating a universal entangling gate operating on Doppler-cooled ions are presented. For ions in a thermal state with n = 18(2), Bell states with a fidelity of0.984(2) were obtained.

The ability to implement high fidelity multi-qubit operations on Doppler-cooled ions is of prac-tical interest in ion trap quantum information processing as the implementation of quantum algo-rithms demands several techniques that do not conserve the ions’ vibrational quantum state: (i)State detection of ancilla qubits as required by quantum error correction schemes [138] can excitethe ion string to a thermal motional state close to the Doppler limit because of the interaction withthe laser inducing the ions to fluoresce. (ii) Experiments with segmented trap structures whereion strings are split into smaller strings also tend to heat up the ions slightly [139]. Here, theavailability of high-fidelity gate operations even for thermal states may provide a viable alternativeto the technically involved re-cooling techniques using a different ion species [140, 141].

Note that all experiments, except the measurements shown in figure 6.4 demonstrating the high

57

Entangling calcium ions

(b)(a)

pulseduration

Time

Lig

ht

po

we

r

slopeduration

Figure 6.1.: Effect of amplitude pulse shaping on non-resonant population transfer causedby a bichromatic light field non-resonantly exciting the carrier transition. Experimentalresults are presented for a gate duration of tgate=25 µs. A comparison of the evolution ofthe populations p2(N), p1(¦), p0(•) for a square-shaped pulse (a) with an amplitude-shapedpulse (b) shows a suppression of the strong non-resonant oscillations for the latter case. Theslopes are shaped as a Blackman window with a duration of 2.5 µs, the figure inset showingthe definitions of pulse and slope duration. Numerical simulations suggest that the actualpulse shape is not so important as long as the switching occurs sufficiently slowly. The solidlines are calculated from (3.34) and (3.29). To match experimental data and simulations,we allowed for a time offset ∆t = 0.5 µs that accounts for the finite switching time of theAOM controlling the laser power.

fidelity gate on thermal ions, use the S1/2, mj = 1/2 level together with the D5/2,mj = 5/2 as aqubit instead of D5/2,mj = 3/2. The choice of the qubits was given by the beam geometry andthe requirement of a high coupling strength (see chapter 3.3.1). The center of mass trap frequencyfor all experiments, except the measurements shown in figure 6.4 (c) (d) , was set to ωax/(2π) =1.232 MHz which corresponds to a Lamb-Dicke parameter η = 0.044. For the mentioned exceptionthe trap frequency was slightly increased to ωax/(2π) = 1.465 MHz which corresponds to a Lamb-Dicke η = 0.04. The main findings of this chapter were also published in [76].

Two 40Ca+ ions were used to perform the experiments in this section. Doppler cooling andfrequency-resolved optical pumping was applied to initialize the ions to |↓↓〉. Then the MS inter-action is switched on and the appropriate parameters are varied to do the measurements.

6.1.1. Amplitude pulse shaping

The merits of amplitude pulse shaping were studied by observing the time evolution of the pop-ulations pk at the beginning of the gate operation when the population transfer is dominated byfast non-resonant coupling to the carrier. For a better visibility of the effect the ions were cooledto the ground state. Figure 6.1 (a) shows the population evolution for the first 5 µs of a 25 µs gateoperation based on a rectangular pulse shape. Averaging over a randomly varying phase ζ1, strongoscillations with a period of 2π/δ=0.84 µs were observed. The phase ζ, defined in chapter 3.5,determines whether the gate operation starts in a maximum or a minimum of the intensity of

1ζ determines whether the gate operation starts in a maximum (ζ = 0) or a minimum (ζ = π/2) of the intensityof the amplitude-modulated beam (see chapter 3.5).

58

6.1 High fidelity two ion Bell states

(b)(a)

Figure 6.2.: (a) Populations p0(N), p1(¦), p2(•) after a single gate operation (tgate=25 µs)where the global frequency detuning of the bichromatic entangling pulse is varied by scanningAO 2. A maximally entangled state is achieved for a global frequency detuning of (2π) 37 kHzrelative to the qubit transition frequency due to AC-Stark shifts. (b) Introduction of a beamimbalance ξ = 0.08 shifts the pattern of the populations by the required amount to fullycompensate for the AC-Stark-shift (note the different x-axis offsets in (a) and (b)). Thesolid lines are calculated by solving the Schrodinger equation for the Hamiltonian given in(3.27) amended by a term accounting for the measured AC-Stark shift.

the amplitude modulated beam. Panel (b) shows that the non-resonant excitations vanish com-pletely after application of amplitude pulse shaping with a slope duration of 2.5 µs correspondingto three vibrational periods of the center-of-mass mode. The slopes were shaped as a Blackmanwindow [142], where the form of the shape is chosen such that a shaped and a rectangular pulseof the same duration have the same pulse area (see inset of panel (b)). Different pulse lengthsare achieved by varying the duration of the central time interval during which the laser power isconstant. The solid lines in the figure are calculated from (3.34) and (3.29).

6.1.2. AC-Stark shift compensation

The AC-Stark shift caused by bichromatic light with spectral components each having a Rabifrequency of Ω/(2π)=220 kHz (for tgate=25µs) is measured by scanning the global laser frequencyusing AO 2 (see figure 4.7). The resulting populations after a gate operation, again for groundstate cooled ions, are depicted in figure 6.2 (a). We observe a drop of the population p1 to zeroat a detuning of (2π) 37 kHz from the carrier transition. At this setting the ions are maximallyentangled. By changing the relative power of the bichromatic field’s frequency components suchthat ξ = 0.08 (see chapter 3.5.1) the AC-Stark shift is compensated. This translates the wholeexcitation pattern in frequency space as can be seen in figure 6.2 (b).

A more sensitive method to infer the remaining AC-Stark shift δAC after a coarse pre-compensationconsists in concatenating two gates separated by a waiting time τw in a pulse sequence akin to aRamsey-type experiment [143] and scanning τw. This procedure maps δAC to a phase φ = δACτw

which is converted into a population change p2 = cos2(φ), p0 = sin2(φ) by the second gate pulse.For two ions, the corresponding Ramsey pattern displayed in figure 6.3 shows oscillations of thepopulations p0 and p2 with a frequency of two times the remaining AC-Stark shift.

59


Waiting time (µs)tw

Figure 6.3.: Population evolution of |↑↑〉 (•) and |↓↓〉 (N) when scanning the waiting timebetween two 25 µs gate pulses in a Ramsey-like experiment. For this scan the detuning ε wasset to (2π) 40 kHz. In this set of data, the AC-Stark shift was only partially compensated byimbalancing the power of the two frequency components. From the sinusodial fits shown assolid lines, we infer an oscillation period of 258(4) µs corresponding to a residual AC-Starkshift of (2π) 1.94(3) kHz.

6.1.3. Gate analysis

A full characterization of the gate operation could be achieved by quantum process tomography [50].At present, however, the errors introduced by the tomography pulse and individual qubit detectionare on the few percent level in our experimental setup which renders the detection of small errorsdifficult in the entangling operation. Instead, the quality of the gate operation was characterizedby using it for creating different Bell states and determining their fidelities.

For the Bell state Ψ1 = |↓↓〉 + i |↑↑〉, the fidelity is given by F = 〈Ψ1| ρexp |Ψ1〉 = (ρexp↑↑,↑↑ +

ρexp↓↓,↓↓)/2 + Imρexp

↓↓,↑↑, with the density matrix ρexp describing the experimentally produced state.To determine F , the populations p2 + p0 need to be measured at the end of the gate operation aswell as the off-diagonal matrix-element ρexp

↓↓,↑↑. To determine the latter, a π/2 pulse was applied toboth ions with an optical phase φ to measure 〈σ(1)

z σ(2)z 〉 for the resulting state as a function of φ.

This procedure is equivalent to measuring oscillations of the expectation value Tr(P (φ)ρexp) of theparity operator P (φ) = σ

(1)φ σ

(2)φ where σφ = σx cos φ + σy sinφ (see figure 6.4 (b) and (d)). The

amplitude A of these oscillations equals 2|ρexp↓↓,↑↑| and is obtained by fitting them with the function

Pfit(φ) = A sin(2φ + φ0).

Measurements [40] using |↓↓〉 as input state have demonstrated Bell state fidelities as high as0.993(1) (see figure 6.4 (a) and (b)) for gate times of 50 µs or 61 trap oscillation periods. Figure6.4 (a) illustrates the population evolution induced by the gate pulse for ground-state cooledions initially prepared in the qubit states |↓↓〉. Figure 6.4 (b) displays parity oscillations for theproduced Bell state. By doubling the detuning to ε/(2π)=40 kHz the gate duration is reducedto only 31 trap oscillation periods and Bell states with a fidelity of 0.971(2) were observed whichis remarkable considering the small Lamb-Dicke parameter of η = 0.044. The detrimental effectsillustrated in figure 6.1 (a) are sufficiently suppressed by amplitude pulse shaping. While manytheoretical papers discussing Mølmer-Sørensen and conditional phase gates put much emphasis on

60


Figure 6.4.: Measured population evolution for p0(•), p1(¦), p2(N) and parity oscillationswith (a,b) and without (c,d) ground state cooling. In the latter case, population is trans-ferred faster into |↑↓, n〉, |↓↑, n〉 as compared to sideband cooled ions due to the highercoupling strength to the sidebands. In (c), the solid lines are a fit to the data points us-ing (3.36) with the mean phonon number n as a free parameter giving n = 18(2). The parityoscillations for the ions in a thermal state of motion have an amplitude of 0.980(2). Com-bining this measurement with the independently determined populations p2 + p0 = 0.988(1)results in a Bell state fidelity of 0.984(2). The data appearing in (a) and (b) are takenfrom [40]. Here the deviation of the dashed lines from the data is caused by the AC-Starkshift compensation using ξ = 0.08. The dashed lines are calculated for ncom = 0.05 from thepropagator (3.29), neglecting pulse shaping and non-resonant carrier excitation. The solidlines are obtained from numerically solving the Schrodinger equation for time-dependentΩ(t) and imbalanced Rabi frequencies ξ = 0.08. The parity oscillations for the ground statecooled ions have an amplitude of 0.990(1) which leads to a fidelity of 0.993(1). The ampli-tude of the parity oscillations was obtained by a fit with the function P (φ) = A ·sin(2φ+φ0).The value of the phase φ0 is without significance. It arises from phase locking the frequenciesν, ν+, ν− and could have been adjusted to zero.

61


the possibility of entangling ions irrespective of their motional state by using these gates, therehas not been any experimental demonstration of this gate property up to now. The reason for thisis that independence of the motional state, as predicted by (3.29), is achieved only deep withinthe Lamb-Dicke regime whereas experiments demonstrating entangling gates on hyperfine qubitsusually have Lamb-Dicke factors on the order of η = 0.1−0.2 [34, 109, 144]. Therefore, all previousexperimental gate realizations used laser cooling to prepare at least the motional mode mediatingthe gate in its ground state with n = 0.

The corresponding time evolution and parity oscillations for ions that are merely Doppler-cooledto a thermal state are shown in figure 6.4 (c) and (d) respectively. As the coupling strengths onthe upper and lower motional sidebands scale as ∝ √

n + 1 and ∝ √n, non-resonant sideband

excitation transfers population much faster from |↓↓, n〉 into |↓↑, n± 1〉, |↑↓, n± 1〉 as comparedto the case of ions prepared in the ground state with n = 0. After the gate time tgate=50 µs,however, the undesired population p1 nearly vanishes as in the case of ground-state cooled ionsand the Bell state Ψ1 is again created. In the experiment, we find a population p1 = 0.012(1) inthe undesired energy eigenstates. The parity oscillations have an amplitude of 0.980(2), resultingin a Bell state fidelity of 0.984(2). The reasons for the somewhat reduced fidelity as comparedto ground-state cooled ions are only partially understood. In part, the fidelity loss arises froma variation of the coupling strength on the vibrational sidebands as a function of n caused byhigher-order terms in η. Fitting equations (3.36) to the population evolution data allows us todetermine the mean vibrational quantum number as n = 18(2). This value is consistent withindependent measurements obtained by comparing the time evolution of the ions when excitingthem on the carrier and on the blue motional sideband. For a thermal state with n = 18 andη = 0.04 calculations show that this effect amounts to additional errors of 5.5×10−3, which makesup for most of the difference.2

The amplitude of the parity oscillations was obtained by a fit with the function P (φ) = A ·sin(2φ + φ0). The value of the phase φ0 is without significance. It arises from phase locking thefrequencies ν, ν+, ν− and could have been adjusted to zero by changing the phase of the signalgenerators producing ν+ and ν−.

As mentioned in Section 3.5.1, the AC-Stark compensation by unbalancing the power of thered and blue frequency component is not applicable to ions in a thermal state. Instead, the laserfrequency needs to be adjusted to account for AC-Stark shifts δAC, a technique that works wellas long as the AC-Stark shifts are smaller than the coupling strength λ of the gate interactionappearing in (3.30) (otherwise, in the case δAC À λ, small laser power fluctuations give rise tolarge phase shifts). Therefore, care must be taken to choose the direction and polarization of thegate laser such that a favorable ratio λ/δAC is obtained. In experiments with a gate durationof tgate=50 µs on the transition S1/2,mj = 1/2 → D5/2,mj = 5/2, we achieved λ/δAC ≈ 3and needed to shift the laser frequency by about 7.5 kHz for optimal Bell state fidelity. In theexperiments shown in figure 6.4, a further reduction of the AC-Stark shift could be obtained byusing a σ+-polarized laser beam incident on the ions along the direction of the magnetic field. Inthis geometry the AC-Stark shift is predominantly caused by the S1/2 ↔ P3/2 dipole transition

2The loss of fidelity due to coupling strength fluctuations is given by 1 − F = (π/2)2(δΩ/Ω)2. The expectationvalue for δΩ/Ω can be estimated by assuming a thermal state distribution and a coupling strength that dependson n in first order as Ω = 1− η2n.

62


Figure 6.5.: Bell state fidelities after n gate operations applied to the input states |↓↓〉(N)and |↓↑〉(•) for tgate = 50µs. Taking into account the error for state preparation of theinput states |↓↑〉 and a similar error to measure the parity signal, we conclude that the gateoperation works on all tested input states similarly well. The solid lines reflect a Gaussiandecay of the parity fringe amplitudes as a function of the number of gates and a linear decayin the desired populations caused by the spectral impurity of the laser. For both input statesthe gate operation implies errors of less than 0.2 after 21 consecutive applications.

since the D5/2,mj = +3/2 state does not couple to any of the 4p Zeeman states. Measurementsshow that the shift is reduced to about 1.8 kHz without compromising the gate speed.

6.1.4. Different input states and multiple gate operations

For a gate time of 50 µs the analysis was extended by applying the gate to the state |↓↑〉 whichis prepared by a π/2 rotation (beam 2) of both ions, followed by a π phase-shift pulse on a singleion performed with the far-detuned focused beam (as explained in chapter 5.4), and another π/2rotation applied to both ions. This pulse sequence realizes the mapping

|↓↓〉 −→ |↓ + ↑〉|↓ + ↑〉 −→ |↓ − ↑〉|↓ + ↑〉 −→ |↓↑〉 (6.1)

to the desired input state for the gate. Imperfections of single-ion addressing led to an error in statepreparation of 0.036(3) (Addressing was improved after these experiments had been performed tothe values stated in chapter 5.4). For the Bell state analysis, the population p1 was measured toinfer ρexp

↑↓,↑↓ + ρexp↓↑,↓↑. Unfortunately parity oscillations cannot be introduced by a collective π/2

pulse acting on the state |↑↓〉 + i |↓↑〉. Instead, this state was transferred into |↑↑〉 + i |↓↓〉 byrepeating the steps of sequence (6.1) as for the state preparation. The coherence was measuredagain by performing parity oscillations.

Figure 6.5 shows a comparison of the fidelity of the gate starting either in |↓↓〉 or |↓↑〉. Thefidelity of a Bell state created by a single gate starting in |↓↑〉 is 0.95(1). Taking into account theerrors for state preparation and the Bell state analysis we conclude that the entangling operationworks equally well for |↓↑〉 as an input state. This hypothesis is supported by the observation thatfor both states we obtain a similar decay of Bell state fidelities with increasing gate number. The

63


Figure 6.6.: Fidelity as a function of the global frequency detuning of the bichromatic lightpulse from the carrier transition (here, the sideband detuning was set to ε/(2π)=20 kHz). Amaximum fidelity of 0.995(4) was found for a detuning of 500 Hz from the transition centerdue to a residual AC-Stark shift. The solid line is obtained by numerically solving equation(3.27) and taking into account the AC-Stark shift compensation by different powers of theblue and the red laser frequency component. At the maximum the solid line experiences asecond order frequency dependence of −9.6(3)× 10−9 Hz2.

application of N gate operations maps the input state |↓↓〉 to

|↓↓〉 −→ |↓↓〉+ i |↑↑〉 −→ |↑↑〉 −→ |↓↓〉+ i |↑↑〉 −→ |↑↑〉 −→ . . .

and the state |↓↑〉 to

|↓↑〉 −→ |↓↑〉 − i |↑↓〉 −→ |↑↓〉 −→ |↑↓〉 − i |↓↑〉 −→ |↓↑〉 −→ . . .

Compared with earlier results [40] where multiple gate operations were induced by varying theduration of a single bichromatic pulse, here up to 21 individual amplitude-shaped pulses wereapplied. Splitting up a long pulse into many shorter gate pulses has no detectable effect on thefidelity of the Bell states produced, and in both cases we obtain a Bell state fidelity larger than0.80 after 21 gates.

6.1.5. Gate errors

From the preceding measurements the following error sources can be determined and quantified(see also [40, 76]).

• A bichromatic force with time-dependent Ω(t) acting on ions prepared in an eigenstate ofSy creates coherent states α(t) following trajectories in phase space that generally do notclose [145, 146]. For the short rise times used in our experiments, this effect can be madenegligibly (< 10−4) small by slightly increasing the gate time.

• Spin flips induced by incoherent off-resonant light of the bichromatic laser field reduce thegate fidelity. The spectrum of the laser determined by a beat measurement (see figure 4.8(b))

64


shows that a fraction γ of about 2·10−7 of the total laser power is contained in a 20 kHz band-width B around the carrier transition when the laser is tuned close to a motional sideband. Asimple model predicts spin flips to cause a gate error with probability pflip = (πγ|ε|)/(2η2B).This would correspond to a probability pflip = 8 · 10−4 whereas the state populations mea-sured for figure 6.5 are consistent with pflip = 2 · 10−3. Spin flip errors could be furtherreduced by two orders of magnitude by spectrally filtering the laser light and increasing thetrap frequency ω/(2π) to above 2 MHz where noise caused by the laser frequency stabilizationis much reduced.

• Imperfections due to low frequency noise randomly shifting the laser frequency νL withrespect to the atomic transition frequency ν were estimated from Ramsey measurementson a single ion showing that an average frequency deviation ∆(νL−ν)/(2π) = 160 Hz oc-curred. From numerical simulations, one can infer that for a single gate operation thisfrequency uncertainty gives rise to a fidelity loss of 0.25% (an infidelity of 10−4 would require∆(νL−ν)/(2π) = 30 Hz). In the parity oscillation experiments shown in figures 6.4 and 6.5,however, this loss is not directly observable since a small error in the frequency of the bichro-matic laser beam carrying out the gate operation is correlated with a similar frequency errorof the carrier (π

2 )φ pulse probing the entanglement produced by the gate so that the phaseφ of the analysing pulse with respect to the qubit state remains well defined.

• Variations in the coupling strength δΩ induced by low-frequency laser intensity noise and ther-mally occupied radial modes were inferred from an independent measurement by recordingthe amplitude decay of carrier oscillations. Assuming a Gaussian decay, a relative variationof δΩ/Ω = 1.4(1) · 10−2 is found. For m entangling gate operations, the loss of fidelity isapproximately given by 1− F = (πm

2 )2(δΩ/Ω)2 and contributes with 5 · 10−4 to the error ofa single gate operation. For the multiple gate operations shown in figure (6.5), this source ofnoise explains the Gaussian decay of fidelity whereas laser frequency noise reduces the fringeamplitude by less than 1% even for 21 gate operations. In combination with error estimatesfor state preparation, detection and laser noise, the analysis of multiple gates provides uswith a good understanding of the most important sources of gate infidelity.

• An error that was not investigated in [40] is the dependence of the Bell state fidelity on theglobal laser frequency detuning from the qubit transition frequency. Experimental results areshown in figure 6.6. The solid line fitting the data is calculated by numerically solving thefull Schrodinger equation for different global frequency detunings and evaluating the fidelity.A second order frequency dependence of −9.6(3)×10−9 Hz2 is found from calculations at themaximum point. This suggests that our laser’s typical mean frequency deviation of 160 Hzcontributes with 3× 10−4 to the error budget.

• A further error source arises when the bichromatic beam couples to both ions with differentstrengths. By recording Rabi oscillations simultaneously on the two ions we conclude thatboth ions experience the same coupling strength Ω to within 4%. From numerical calculationswe infer an additional error in the measured Bell state fidelity of less then 1× 10−4.

• Another possible error source is heating of the COM-mode during the gate operation since

65


the gate is not insensitive to motional heating in the parameter regime of our implementation.Using the calculation performed in [81], we find a fidelity reduction of ∆F = Γhtgate/2 whereΓh is the heating rate of the COM-mode. As in our experiments Γh = 3s−1, the fidelity isreduced by ∆F ≈ 10−4 for tgate=50 µs.

6.1.6. Conclusion

Until recently, entangling gates for optical qubits were exclusively of the Cirac-Zoller type whichrequire individual addressing of the ions. Compared to this type of gate the Mølmer-Sørensengate gives an improvement in fidelity and speed of nearly an order of magnitude. The achievedfidelity sets a record for creating two-qubit entanglement on demand irrespective of the physicalrealization considered so far. The results with concatenations of 21 of these operations bring therealization of more complex algorithms a step closer to reality. The implementation of a gatewithout the need for ground state cooling is of particular interest in view of quantum algorithmsthat require entangling gates conditioned on quantum state measurements that do not preservethe ions’ motional quantum state.

When considering gate imperfections, two regimes are of interest: On the one hand, in view ofa future implementation of fault-tolerant gate operations, it is important to investigate whetherthe gate operation allows in principle for gate errors on the order of 10−4 or below. On the otherhand, for current experiments aiming at demonstrating certain aspects of quantum informationprocessing, errors on the order of 10−2 are not forbiddingly high. For these experiments, theprospect of carrying out a gate operation using ions that are not in the vibrational ground stateof the mode mediating the internal-state entanglement, is appealing as it might allow to performentangling gates after having split a long ion string into shorter segments (the splitting process hasbeen demonstrated to heat up the ions by no more than a single quantum of motion [42]). Similarly,quantum state detection by light scattering on a cycling transition heats up the vibrational mode.However, if done properly, the final mean quantum number stays well below the average of 20quanta for which we demonstrated entanglement generation. Therefore, experiments involvinggates after splitting and detection operations might profit from a quantum gate as demonstratedin chapter 6.1.

For future ion trap experiments in the fault-tolerant domain, the needs are going to be different.Here, ground state laser cooling will most likely be indispensable to achieve the highest fidelitypossible. Also AC-Stark compensation based on imbalanced bichromatic beam intensities shouldbe avoided as the technique tends to complicate the gate Hamiltonian and to introduce smalladditional errors. Even though the current experiments are still limited by technical imperfections,simulations predict that in principle it should be possible to achieve gate errors of 10−4 or belowwith a Mølmer-Sørensen gate on a quadrupole transition. Gates with ions in motional states areimportant in this context as no experiment will cool the ions to the ground state n = 0 perfectly(in current experiments, the ground state is typically occupied with a success rate of 90 to 99%).Our simulations indicate that for ions in n = 1, gate errors could still be as small as 2 · 10−4 sothat gate errors of 10−4 or below seem feasible even without perfect initialization of the motionalmode.

The optical qubit as used here is certainly not the best solution for long time storage of quantum

66

6.2 Three ion entangled states and arbitrary operations on three qubits

Figure 6.7.: (a) Evolution of the populations off three ions p0(O), p1(•), p2(M) and p3(¨)induced by a Mølmer-Sørensen interaction of pulse length τ . Pulse shaping is used tosuppress off-resonant excitations. At 100 µs the maximally entangled state |↓↓↓〉 − |↓↑↑〉 −|↑↓↑〉 − |↑↑↓〉 is created. The solid lines are calculated from the propagator (3.29). (b)Populations p0(O), p1(•), p2(M) and p3(¨) after a single gate operation (tgate=100 µs) wherethe global frequency detuning of the bichromatic entangling pulse is varied by scanningAO 2. A maximally entangled state is achieved at a detuning of 700 Hz relative to the qubittransition frequency due to uncompensated AC-Stark shifts. The solid lines are calculatedby solving the Schrodinger equation for the Hamiltonian given in (3.27) amended by a termaccounting for the measured AC-Stark shift.

information. Instead, qubits encoded in two hyperfine ground states whose frequency difference isinsensitive to changes in magnetic field are preferable. These magnetic-field insensitive hyperfinequbits can store quantum information for times exceeding the duration of the gate operationpresented here by more than four orders of magnitude [102, 147, 148]. However, on such qubitstates no high-fidelity universal gates have been demonstrated so far. By mapping between thehyperfine qubit encoded in the ion’s ground states and the optical qubit we will benefit from bothof their advantages (see chapter 6.3).

6.2. Three ion entangled states and arbitrary operations on

three qubits

After entangling two ions with a MS interaction the next obvious step is to entangle more ions.Unfortunately in the current setup we are not able to reliably trap more than three ions in acrystal-like structure with reasonable trapping parameters. The issue is not yet clear, but mostlikely this is due to a high collision rate of the ions with background gas particles. An indication forthis is, that on average every 4 h an ion in a crystal becomes dark due to a chemical reaction withthe background gas. Measurements of the axial oscillation frequency of a two ion crystal with onedark ion indicated that an OH group attached itself to a dark ion. As we are not able to trap morethan four or more ions the experiments presented in this chapter only deal with entangling threeions. The results are very similar to the ones recently published by Chris Monroe’s group [149].The difference is that in this thesis the qubit is an optical one and a single axial oscillator mode isused for mediating the coupling.

67


Figure 6.8.: (a) A (π2 )(−y) pulse applied to all ions maps the three ion entangled state

|↓↓↓〉−|↓↑↑〉−|↑↓↑〉−|↑↑↓〉 onto the GHZ state |↓↓↓〉+ i |↑↑↑〉. A second(

π2

)φ

analysis pulseagain applied to all ions gives rise to parity oscillations P (φ) = A · sin(3φ) as a function of φ.The obvious difference to parity flops with two ions is the different period, three oscillationsoccurring for φ changing from 0 to 2 π. A fit with the function P (φ) = A · sin(3φ + φ0)yields the parity fringe amplitude A = 0.977(2). The value of the phase φ0 is withoutsignificance. Together with the independently measured populations p1 + p2 = 0.0189(1)this leads to a GHZ state fidelity of 0.979(2). (b) Parity like oscillations of the states(|↑↑〉1,2 + i |↓↓〉1,2)⊗ |↓〉3 () and (|↑↓〉1,2 + |↓↑〉1,2)⊗ |↓〉3 (♦) created by entangling two outof three ions. The analysis

(π2

)φ

pulse was applied to two ions and the state evolution wasanalyzed with the camera. The addressing of all operations on two ions is clearly visiblein the states |↑↑〉1,2 ⊗ |↑〉3 , i |↓↓〉1,2 ⊗ |↑〉3 (combined in •) and |↑↓〉1,2 ⊗ |↑〉3 , |↓↑〉1,2 ⊗ |↑〉3(combined in N) as they show no oscillatory behavior and are not populated. From thesedata one can infer a state fidelity of 0.958(2) for entangling two ions and leaving the thirdone untouched.

6.2.1. A three ion Mølmer-Sørensen gate

Three 40Ca+ ions were loaded into the linear trap. After Doppler cooling and frequency-resolvedoptical pumping, the axial center of mass mode is cooled close to the motional ground state.The ions are now initialized to |↓↓↓〉. Then, the gate operation is performed with beam 1 (seefigure 5.4)(a) and the probabilities pk of finding k ions in the state |↓〉 are measured for a varyingpulse duration. A Rabi frequency of Ω/(2π) ≈ 70 kHz is required for performing a gate operationwith ε/(2π) = 10 kHz and a three-ion Lamb Dicke parameter of η = 0.036. To make the bichromaticlaser pulses independent of the phase ζ, the pulse is switched on and off by using pulse slopes of2 µs duration.

The time evolution of the three-ion state populations pk are shown in figure 6.7 (a). Aftera gate time of tgate=100 µs the undesired populations p0 and p2 nearly vanish and the state|↓↓↓〉−|↓↑↑〉−|↑↓↑〉−|↑↑↓〉 is created. In the experiment, we find a population p0 +p2 = 0.0189(1)in the undesired energy eigenstates. Contrary to the two ion case, an application of two times thegate does not flip the state of all qubits but rather returns all populations to the initial state.

The AC-Stark shift caused by the bichromatic light is not compensated. A scan of the globallaser frequency reveals a vanishing of the undesired populations p0 +p2 at 700 Hz detuning relativeto the carrier transition. The resulting frequency dependence of the populations after a gateoperation are depicted in figure 6.7 (b).

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6.2 Three ion entangled states and arbitrary operations on three qubits

For the state Ψ1 = |↓↓↓〉−|↓↑↑〉−|↑↓↑〉−|↑↑↓〉, the fidelity is not directly observable with parityoscillations. First a Ux(π

2 ) pulse is applied to all ions mapping the entangled state |↓↓↓〉 − |↓↑↑〉 −|↑↓↑〉 − |↑↑↓〉 onto the GHZ state |↓↓↓〉 − i |↑↑↑〉. A second U(π

2 , φ) analysis pulse again applied toall ions gives rise to parity oscillations P (φ) = A ·sin(3φ) as a function of φ (see figure 6.8 (a)). Theobvious difference to parity flops with two ions is the different period, three oscillations occurringfor φ changing from 0 to 2 π. A fit with the function P (φ) = A · sin(3φ + φ0) yields the parityfringe amplitude A = 0.977(2). Combining this measurement with the previously measured statepopulations a fidelity of 0.979(2) is obtained for Ψ1.

6.2.2. Entangling two out of three ions

The goal for all QIP experiments is to implement arbitrary quantum circuits. Here a first step to-wards that direction is presented by combining a global MS-interaction with addressed σz rotationsand global carrier pulses as described in chapter 5.4.

The most important operation is to implement the pulse sequence shown in figure 5.4(c). Thispulses sequence effectively switches off the interaction between one ion and the other two. Afterthe application of these pulses we end up in the state Ψ2 = (|↑↑〉1,2 + i |↓↓〉1,2) ⊗ |↓〉3 where thesubscript denotes the ion. To analyze the fidelity of this state we again apply a

(π2

)φ

pulse tothe two entangled ions by using the same principle. All ions are first rotated by

(π4

)φ, then ion

1 experiences a π phase shift by an AC-stark pulse. The second(

π4

)φ

pulse rotates ion 2 and 3by the desired angle and returns ion 1 to the initial state. This pulse sequence induces parity-likeoscillations between the states Ψ2 and (|↑↓〉1,2 + |↓↑〉1,2)⊗ |↓〉3. The state of the ions was detectedwith the help of the camera and the recorded oscillations are shown in figure 6.8. The addressing ofall operations on two ions is clearly visible in the evolution of the states |↑↑〉1,2⊗|↑〉3 , |↓↓〉1,2⊗|↑〉3and |↑↓〉1,2 ⊗ |↑〉3 , |↓↑〉1,2 ⊗ |↑〉3. These states are not populated, thus they effectively did notparticipate in any of the operations. From these data combined with the independently measuredpopulation one can infer a state fidelity of 0.958(2) for the state Ψ2.

Realizing this pulse sequence opens up the possibility for more complicated sequences as allthe necessary building blocks have been demonstrated. Recent experiments published by thegroup of David Wineland demonstrated a different approach to implement arbitrary quantumcircuits [55, 56]. They “designed“ their interactions by moving the ions in a segmented trap andre-cooling them with a different ion species.

One particularly interesting algorithm is the one for correcting phase errors on a single qubit asshown in figure 6.9. The first MS gate encodes qubit 1 in a three ion state that is protected againstphase errors. After the error occurred, the following pulses detect and correct for the error. Afterthe operation only ion 1 is in the corrected state, ion 2 and 3 are in some arbitrary state. Ion1 can be again protected by resetting ion 2 and 3 to |↓↓〉 and applying another entangling pulse.The phase errors that can be corrected are either a phase flip on one ion or small phase errors onall of them, e.g. a π/10 phase error on all of them is corrected in more than 99.8% of the casesassuming perfect operations. In this sequence one can identify a building block very similar to theone entangling two out of three ions. Meanwhile experiments have been performed implementingthis algorithm and will be published soon. Similar algorithms using the same NMR-like techniquesand building blocks were investigated by Volckmar Nebendahl in his thesis [150]. The sequence for

69


Figure 6.9.: Error correction sequence for three ions. The first MS gate encodes qubit 1in a three ion state. After this encoding a phase error can occur and the subsequent pulsesdetect and correct for it. After the operation only ion 1 is in the corrected state, ion 2 and3 are in some arbitrary state. Ion 1 can be again protected against phase errors by resettingion 2 and 3 to |↓↓〉 and applying another entangling pulse. The phase errors that can becorrected are either a phase flip on one ion or smaller phase errors on all of them. The greybox corresponds to the block already implemented, entangling two out of three ions.

the error correction code shown here and multi qubit gates like a Toffoli gate [151] can be foundin his thesis.

6.3. High fidelity entanglement of 43Ca+ hyperfine clock states

In experiments with trapped ions most of the elementary building blocks for quantum informationprocessing have been demonstrated. State initialization, long quantum information storage times,entangling gates and readout have been realized with high fidelity [40, 109, 147, 148, 152, 153].A major challenge in current experiments is to integrate these building blocks into a single setupand make them work for a given ion species and parameter range set by the trap frequencies, themagnetic field strength and further parameters. Quantum information encoded in ground stateatomic levels whose energy difference only weakly depends on changes of the magnetic field (“clock“states) has been stored for more than a second [147, 148, 154, 155].

Most high fidelity entangling operations have been demonstrated on qubits with limited coher-ence times [102, 109]. One exception is the entanglement of hyperfine clock states in a Yb+ systemdemonstrated by Kim et al. [149].

An attractive way of combining high fidelity entangling gates with long storage times is tocoherently map the quantum information from clock states to atomic states which are suitable forperforming the entangling operation with high fidelity, and to map the information back at theend of the entangling operation [147]. In case this mapping operation is addressed at individualions - for instance by using a strongly focused laser - the entanglement operation can be appliedto the entire ion string.

The coherence time of the optical qubit used for entangling the ions is limited by magneticfield fluctuations, laser linewidth and ultimately the 1.2 s lifetime of the D5/2 state. For the mostabundant isotope 40Ca+, encoding the qubit in long-lived hyperfine clock states is impossible as thisisotope has a nuclear spin I = 0. In the isotope 43Ca+ on the other hand where I = 7/2, coherencetimes of many seconds have been measured for qubits encoded in the S1/2(F = 4, mF = 0) andS1/2(F = 3, mF = 0) states [154, 155].

70

6.3 High fidelity entanglement of 43Ca+ hyperfine clock states

Figure 6.10.: (a) Energy levels of 43Ca+ showing the hyperfine splitting of the atomicstates S1/2 and D5/2. Microwave radiation applied to an electrode close to the ions drives thehyperfine qubit encoded in the states |↓〉 and |↑〉. A laser at 729 nm excites the ions on thetransition from the S1/2(F = 4) to the D5/2(F = 2, .., 6)-states. It is used for ground statecooling, state initialization, state discrimination and to excite the optical qubit comprisedof the states |↓〉 and |⇑〉 or |⇑’〉. (b) Using a bichromatic laser beam, two 43Ca+ ions areentangled on an optical transition with a maximum target state fidelity of 96.9(3)% at amagnetic field of 6 G. By introducing a waiting time between the creation and the analysisof the Bell state we observe a Gaussian decay of the Bell state fidelity (•) (black solid line).Data for the first 3 ms are displayed as inset. When the optical qubit is mapped to thehyperfine qubit subsequent to the gate operation we obtain a maximum Bell state fidelityof 96.7(3)% and the lifetime of the entanglement increases to 96(3) ms (F). Measurementstaken at a magnetic field of 3.4 G give similar results (N) for the coherence time. Here, theoptical qubit |↓〉 ↔ D5/2(F = 4,mF = 2) in combination with the laser beam 3 was used forentangling the ions. Each data point represents 18.000 to 24.000 individual measurements.

In this chapter a combination of the long quantum information storage times observed for thehyperfine qubit of a single 43Ca+ ion with the high fidelity gate operation on optical qubits isinvestigated by mapping between these two qubits. The steps for state initialization and Bellstate preparation in the optical qubit are discussed. Then the mapping to the hyperfine qubit isdescribed and for both qubits the entanglement decay is measured. Remapping to the optical qubitafter a 20 ms storage time and the application of further gate operations is demonstrated. Dueto the nuclear spin of I = 7/2, the isotope 43Ca+ has a relatively complex energy level structurecompared to other ions used for quantum information processing. Magnetic field and polarizationsettings are discussed to reduce errors due to close by Zeeman-levels, which are especially harmfulfor gate operation in the case when large coupling strengths are required in a relatively shorttime. The findings of this chapter were also published in reference [77]. Similar techniques wereimplemented in experiments conducted in David Wineland’s group [55, 56].

6.3.1. Qubit preparation and manipulation

An energy level scheme for 43Ca+ with all relevant hyperfine levels is shown in figure 6.10 (a). In theexperiments reported here, the hyperfine qubit, comprised of the states |↑〉 ≡ S1/2(F = 3,mF = 0)and |↓〉 ≡ S1/2(F = 4,mF = 0), is driven by applying a microwave field (3.2 GHz) to one ofthe electrodes which is also used to compensate for external electric stray fields (see figure 4.1

71


electrodes (C)).

Each experimental sequence starts by Doppler-cooling the ions for 3 ms followed by sidebandcooling of both axial modes close to the ground state (ncom, nst ≈ 0.03(3), 0.4(1); 5 ms and 3 ms).Optical pumping (as described in chapter 5.4) initializes both ions in the energy level S1/2(F =4,mF = 4) in more than 98.9% of the cases.

Subsequently the populations of both ions are transferred by a π-pulse to the energy levelD5/2(F = 3,mF = 2) (transfer 1). By turning on the lasers at 397 nm and 866 nm for 500 µs

one can detect residual population remaining in the S1/2 state manifold (PMT check). All caseswhere more than 3 photons are detected are rejected (∼ 20 photons/ms are received per ion inthe S1/2-state at a dark count rate of 0.2 photons per ms). The rejection rate is typically 2%.Initialization of the hyperfine qubit is completed by another π-pulse (transfer 2) on the transitionD5/2(F = 3,mF = 2) ↔ S1/2(F = 4,mF = 0) ≡ |↓〉. In total the initialization of internal andexternal degrees of freedom takes about 12 ms and the qubit state |↓↓〉 is populated with a fidelityof more than 99.2%. The fidelity of state initialization to |↓↓〉 is measured by transferring thepopulation with two consecutive π-pulses from |↓↓〉 to two different Zeeman-levels in the D5/2

manifold (D5/2(F = 3,mF = 2) and D5/2(F = 5,mF = 2)) and determining the remainingpopulation in the S1/2 by fluorescence detection. The fidelity of optical pumping is determined ina similar way.

The magnetic field is set to 6.0 G in order to achieve a sufficiently large frequency separation ofthe neighboring transition lines in the spectrum of the S1/2 ↔ D5/2 quadrupole transition [126].The transition |↓〉 ↔ D5/2(F = 6,mF = 1) ≡ |⇑〉 is chosen as optical qubit because of its smallmagnetic field sensitivity of 350 kHz/G (which is at least a factor of two lower than the sensitivityof any mF = 0 to mF = 0 transition for the chosen magnetic field) and in order to avoid havingcoinciding resonances with (micro)motional sidebands of other spectral components. Moreover,this choice of optical qubit has the advantage of a comparatively large Clebsch-Gordan coefficientwhich together with a carefully set laser polarization (for this qubit we use beam 1 as shown infigure 5.4(a)) suppresses neighboring Zeeman transitions with ∆m 6= +1. The transitions ∆m = 0and 2 are suppressed by factors of 38 or more in terms of their Rabi frequencies, the strongestbeing the |↓〉 ↔ D5/2(F = 6,mF = 0) transition, whose resonance is more than 4 MHz away fromthe |↓〉 ↔ |⇑〉 transition. The two closest transitions are S1/2(F = 4,mF = ±1) ↔ |⇑〉 (& 2 MHzaway) which are suppressed by a factor of 270 in coupling strength. The high coupling strengthon the gate transition reduces the required power for the gate operation and thus the AC-Starkshift caused by off-resonant coupling to dipole transitions.

After initialization to |↓↓〉, a Mølmer-Sørensen entangling operation (MS 1) consisting of asingle bichromatic laser pulse is applied to the optical qubit transition to create a Bell state of|↓↓〉 + i |⇑⇑〉. The gate time τgate = 100 µs corresponds to a detuning of 10 kHz from the axialsideband. For the chosen beam geometry, these settings lead to an AC-Stark shift of 3.5 kHz whichis fully compensated by using different coupling strengths Ωb, Ωr for the red and the blue detunedfrequency component with ξ = 0.14.

72

6.3 High fidelity entanglement of 43Ca+ hyperfine clock states

Table 6.1.: Duration and errors of each experimental step and the achieved state fidelities.Experiment Duration Error State

step (µs) (%) fidelity (%)Opt. pumping 600 < 1.1 > 98.9

Transfer 1 20 0.7 98.2PMT check 500 0.7 99.3Transfer 2 20 < 0.1 > 99.2

Total prep. |↓〉 1140 < 0.8 > 99.2MS 1 100 2.3 96.9Map 120 0.2 96.7Wait 20× 103 2.2 94.6

Map−1 120 0.7 93.9MS 2 100 3.7 90.4MS 3 100 2.9 87.8

6.3.2. Experimental results

The fidelity of the created Bell state is determined by measuring the populations p0+p2 in |↓↓〉 and|⇑⇑〉 and the coherence between the two states. The coherence is again inferred from the amplitudeof parity oscillations. The maximum fidelity was determined to 96.9(3)%. The durations and errorsof all experimental steps and the achieved state fidelities are given in table 6.1.

By introducing a waiting time before the Bell state analysis, a Gaussian decay of the parityfringe contrast is observed. From this model a Bell state fidelity of 75% after 3.43(5) ms isextrapolated (see inset of figure 6.10(b)) corresponding to a loss of 50% of the coherence since thestate populations hardly change. The magnetic field sensitivity of this optical qubit is 350 kHz/Gwhich is one cause of dephasing. However, in measurements where a single 43Ca+ ion is probed onthe first order field insensitive transition (S1/2(F = 4,mF = 4) ↔ D5/2(F = 4, mF = 3) at 3.4 G)a single qubit coherence times of 8.1(3) ms is obtained (see figure 4.8(a) inset). This leads to theconclusion that the decoherence on the optical qubit is also caused by acoustical noise picked upby the 2 m fiber cord used and the finite linewidth (∼ 20 Hz) of the laser.

Mapping the optical qubit to the hyperfine qubit is achieved by a microwave π-pulse on the hy-perfine qubit transition followed by a π-pulse on the optical qubit (map) which results in an averageerror rate of 0.2%. Detection of the hyperfine qubit states is done by shelving the population of |↓〉to the D5/2 manifold by two consecutive π-pulses to different Zeeman states (D5/2(F = 5,mF = 2)and D5/2(F = 6,mF = −2)) followed by fluorescence detection. A Bell state fidelity measurementafter the mapping yields a parity fringe contrast of 95.7% - induced by varying the phase of amicrowave π/2-pulse - corresponding to a target state fidelity of still 96.7%. By delaying the stateanalysis a Gaussian decay of the parity fringe pattern is observed (figure 6.10) corresponding to a50% loss of phase coherence on a time scale of 96(3) ms which means that a factor 28 in coherencetime is gained compared to the optical qubit.

For a successive gate application the Bell state is mapped back to the optical qubit (map−1)after a waiting time of 20 ms and the gate (MS 2) is applied a second time, disentangling the ionsto |⇑⇑〉 in 90.4% of the cases, indicating an error of 3.7%. Another subsequent application of thegate (MS 3) adds an error of 2.9% leading to |↓↓〉 − i |⇑⇑〉 with a fidelity of 87.8%.

The results with successive gates suggest that the Bell state fidelity depends slightly on whether

73


the gate is applied to the input state |↓↓〉 or |⇑⇑〉, an effect already observed in previous experiments.This observation can be explained by the different spectator states into which the populationcan leak by unwanted excitations. The effect is most prominent in a series of measurementswhere the magnetic field was set to 3.4 G. For this field strength the Zeeman splitting of theground state and the axial trap frequency coincide to within 10 kHz. Using the transition |↓〉 ↔D5/2(F = 4,mF = 2) ≡ |⇑’〉 as optical qubit the polarization of beam 3 (see figure 5.4(a)) isadjusted to suppress the coupling strength to the nearest neighboring transitions. Note that inthis geometry and polarization setting, both ∆m = ±2 have optimal coupling, so a higher gatelaser power is required as compared to |⇑〉. This leads to Bell state fidelities of up to 96.3% aftera single gate operation applied to |↓↓〉 . The decoherence for these Bell states after mapping tothe hyperfine qubit is also shown in figure 6.10(b). Applying the gate operation on the inputstate |⇑’⇑’〉 the maximum obtained Bell state fidelity after a single gate operation is only 92%.Further measurements at a magnetic field of 1.36 G where we expected no coincidences of spectralcomponents reveal a maximum Bell state fidelity of 95%. For these measurements, beam 1 wasused again with |↓〉 ↔ |⇑〉 as the optical qubit.

6.4. Comparison between 40Ca+ and 43Ca+.

Several experiments including this thesis have demonstrated that 40Ca+ and 43Ca+ ions are suitedfor QIP processing. Most DiVincenzo criteria including high fidelity entangling gates have beendemonstrated for both isotopes. 40Ca+ has the advantage of a very simple level structure but lacksthe possibility to encode qubits in states with low magnetic field dependence. Furthermore thecoherence time of the optical qubit suffers from laser frequency noise. Handling 43Ca+ is muchmore difficult due to its complicated level structure. The advantage of 43Ca+ is that qubits can beencoded in hyperfine and optical states with low magnetic field dependence. The hyperfine qubithas an additional advantage as it can be manipulated by either a Raman laser or a microwave fieldsuch that the frequency instability of the driving field is negligible.

Comparing the results for entangling 43Ca+ ion with the results obtained for entangling 40Ca+

ions, one can conclude that low errors of the Mølmer-Sørensen interaction on the optical qubitrequire that the qubit transition is well isolated from other spectral components such that noother transitions are erroneously excited. This view is supported by the observation that the gatemechanism favors a high magnetic field, where the transitions to neighboring lines are spectrallywell separated. However, the sensitivity to magnetic field fluctuations of the hyperfine qubitincreases for higher fields. Therefore, a particularly interesting regime for 43Ca+ ions occurs at amagnetic field of 150 G where nonlinearities in ground state Zeeman splitting lead to a first ordermagnetic field insensitive transition S1/2(F = 4,mF = 0) ↔ S1/2(F = 3, mF = 1). This promiseseven longer coherence times as well as smaller errors for all spectrum dependent operations sincethe Zeeman splitting is huge compared to the measurements presented here.

The following points summarize and compare the operations used to handle the optical qubit in40Ca+ and the optical and hyperfine qubit in 43Ca+. Additionally the coherence times achievedin this experiment for different qubits are given.

74

6.4 Comparison between 40Ca+ and 43Ca+.

Doppler-cooling is more complicated for 43Ca+:

• 40Ca+: One π-polarized 397 nm laser and one 866 nm laser is sufficient for Doppler-cooling.

• 43Ca+: Multiple frequency components in the σ+-polarized 397 nm laser and the 866 nmlaser are necessary for Doppler cooling. Additionally, a second π-polarized 397 nm laser isneeded to avoid population trapping in dark states.

State initialization of the 43Ca+ hyperfine qubit is more complicated than initializing the opticalqubit:

• optical qubit: Optical pumping with a σ+-polarized 397 nm is sufficient to initialize the stateof 40Ca+ to S1/2,m = 1/2 or the state of 43Ca+ to S1/2, F = 4,mF = 4.

• hyperfine qubit: To initialize the state S1/2, F = 4, mF = 0 two additional π pulses on thequadrupole transition or four microwave π pulses are necessary.

The Mølmer-Sørensen gate operation works better for 40Ca+ ions:

• 40Ca+: A Mølmer-Sørensen gate operation has been achieved which generated Bell stateswith fidelities exceeding 99%. [40]

• 43Ca+: The gate fidelity is limited by off-resonant excitation of close-by Zeeman-states toabout 97%. [77]

The coherence properties of 43Ca+ qubits are better than the coherence properties of 40Ca+ qubits:

• 40Ca+ optical qubit: The coherence time for a single ion is limited to about 3 ms by magneticfield fluctuations and laser frequency noise.

• 43Ca+ optical qubit: Due to the availability of magnetic field independent transitions thecoherence time is only limited by the laser stability. For a single ion a coherence time of 8 mswas measured.

• 43Ca+ hyperfine qubit: For the hyperfine qubit S1/2, F = 4, mF = 0 ↔ S1/2, F = 3,mF = 0a coherence time of several seconds has been measured. [154]

Which isotope is chosen for a particular experiment depends on what sets the experimental limitin the sequences that should be run. If the sequence is short and one is interested in the bestoverall fidelity (considering the current experimental status) and the easiest handling, then 40Ca+

should be used. If the limit is set by the coherence time then one should consider using 43Ca+. Itis not absolutely necessary to use the hyperfine qubit in 43Ca+. The coherence time can alreadybe increased by choosing a magnetic field independent transition on the optical qubit in 43Ca+.This has the advantage that the handling is still quite simple and one avoids the complicated readout and initialization steps of the hyperfine qubit.

75


76

7. Experimental test of quantum

contextuality

Since the beginning of quantum mechanics (QM) it has been debated whether hidden variable(HV) theories can account for the intriguing features of QM [64, 156]. Especially the indetermin-ism of QM was a feature some physicists could not accept and was one of the reasons why HVtheories came up. In the sixties, Bell found that local HV theories cannot reproduce the quantummechanical correlations for local measurements on entangled states [65]. Later, a series of exper-iments confirmed a conflict between HV and QM theories by realizing a refined version of Bell’s“Gedankenexperiment“ [66–69]. These experimental refutations led to additional assumptions andrestrictions for the structure of local HV theories.

Kochen, Specker and Bell [70–72] proposed another HV theorem testing for non-contextuality ofmeasurements. Non-contextuality is the property of a HV model that the value of a measurementv(A) is determined, regardless of which other compatible measurement is measured jointly withA. Two or more measurements (A,B,C,. . .) are compatible if they can be measured jointly on thesame individual system without disturbing each other. Compatible measurements can be madesimultaneously or in any order and can be repeated any number of times on the same individualsystem and must always give the same result independently of the initial state of the system.A context is a set of compatible measurements. A physical model is called non contextual if itassigns a measurement result independent of which other compatible measurements are carried outon the probed system. Non contextuality is a plausible assumption for all physical models especialfor measurements on distant systems or in the case of measurements concerning different degreesof freedom on the same system. The KS theorem states that noncontextual HV theories cannotreproduce the predictions of QM.

For some time it has been debated whether the Kochen Specker (KS) theorem could be ex-perimentally tested at all [157, 158]. Nevertheless an experiment has been proposed testing forthese HV theories [159]. Here the first experiment testing a complete Kochen Specker theorem [75]falsifying non-contextual HV theories is presented. Recently additional results with photons [73]and NMR qubits [74] have shown similar results.

In the following chapter the notions compatibility and contextuality as well as the KS theoremwill be discussed. Then the experimental realization of quantum non-demolition measurementsand a test of the Kochen Specker theorem will be shown. At the end of this chapter the results willbe analyzed and possible loopholes will be discussed. The first section is taken from reference [160]whereas the experimental findings were already published in reference [75].

77

Experimental test of quantum contextuality

7.1. The Kochen-Specker theorem

7.1.1. Measurement scenario

The situation in the experiment will be the following: On an individual system a sequence ofdichotomic measurements (outcomes ± 1) will be performed. The question that arises is: underwhich conditions can the result of such measurements be explained by HV theories? Or moreprecisely, which conditions a HV model has to violate to reproduce the QM predictions? The HVmodel will be described by a distribution p(λ) with the hidden variable λ ∈ Λ from a set Λ. Thedistribution summarizes all information about the past, including all preparation steps and allmeasurements already performed. Causality is assumed, so the distribution is independent of anyevent in the future. It rather determines all the probabilities of the results of all possible futuresequences of measurements. We assume that, for a fixed value of the HV, the outcomes of futuresequences of measurements are deterministic, hence all indeterministic behavior stems from theprobability distribution.

7.1.2. Notation

In an experiment the state of the system is prepared in a repeatable way. In the language of HV-models this leads to a probability distribution pexp(λ) that can be repeatedly produced even thoughits actual form is unknown. In a single instance of an experiment, one obtains a state determinedby the HV λexp. Which λexp has been prepared cannot be controlled by the experimentalist andthe HV is distributed according to pexp(λ).

The following notation will be used to describe measurements and their outcomes. Ai denotesthe measurement of the property A at the position i in the sequence. A sequence of measurementsis denoted by A1B2C3 when A is measured first, then B and finally C. The result of a specificmeasurement e.g. B2 in such a sequence is written as v(B2|A1B2C3). In addition v(A1B2C3) =v(A1|A1B2C3)v(B2|A1B2C3)v(C3|A1B2C3). For the probability distribution p(λ), one can defineprobabilities like p(B+

2 |A1B2C3) [or p(B+2 C−3 |A1B2C3)] denoting the probability of obtaining the

value B2 = +1 [and C3 = −1] when the sequence A1B2C3 is measured. Then, one can alsoconsider mean values like 〈B2|A1B2C3〉 = p(B+

2 |A1B2C3)− p(B−2 |A1B2C3), or the mean value of

the complete sequence, 〈A1B2C3〉 = p[v(A1B2C3) = +1]− p[v(A1B2C3) = −1].

7.1.3. Compatibility of measurements

First the notion of compatibility has to be defined. In the simplest case, this is a relation betweena pair of observables, A and B. For that, let SAB denote the (infinite) set of all sequences, whichuse only measurements of A and B, that is, SAB = A1, B1, A1A2, A1B2, B1A2, . . .. Then one canformulate:

Definition 1. Two observables A and B are compatible, denoted by A ∼ B, if the following twoconditions are fulfilled:

(i) For any instance of a state (that is, for any λexp) and for any sequence S ∈ SAB the obtained

78

7.1 The Kochen-Specker theorem

values of A and B remain the same, i.e.

v(Ak|S) = v(Al|S),

v(Bm|S) = v(Bn|S),

where k,l,m,n are all possible indices for which the considered observable is measured at thepositions k,l,m,n in the sequence S.

(ii) For any state preparation (that is for any pexp(λ)), the mean values of A and B during themeasurement of any two sequences S1, S2 ∈ SAB stay the same, i.e.

〈Ak|S1〉 = 〈Al|S2〉 ,〈Bm|S1〉 = 〈Bn|S2〉 .

Conditions (i) and (ii) are necessary conditions for compatible observables. Two observables whichviolate one of them cannot be called compatible.

It is important to note that the compatibility of two observables is experimentally testable bypreparing all possible λexp or pexp(λ). The fact that these sets or the set SAB may be infinite isnot a specific problem here, as any measurement device or physical law can only be tested in afinite number of cases.

One should note, that (ii) does not necessarily follow from (i), as will be illustrated by thefollowing example: Consider a HV model where, for any λ, all v(Ak|S) are +1 when the firstmeasurement in S is A1, while they are −1 when the first measurement is B1. The values v(Bm|S)are always +1. Then, condition (i) is fulfilled, while (ii) is violated, since 〈A〉 = 1 but 〈A2|B1A2〉 =−1.

Finally, it should be added that the notion of compatibility can be extended to three or moreobservables. For instance, if three observables A,B,C are investigated, one considers the set SABC

of all measurement sequences involving measurement of A,B or C and extends conditions (i) and(ii) in an obvious way.

7.1.4. Operational definition of noncontextuality

Noncontextuality means that the value of any observable A does not depend on which othercompatible observables are measured jointly with A. For the model, noncontextuality is formulatedas a condition on a HV model as follows:

Definition 2. Let A,B,C be observables in a HV model, where A is compatible with B, and A isalso compatible with C: The HV model is noncontextual if it assigns, for any λ an outcome of Awhich is independent of whether B or C is measured before or after A, that is,

v(A1|A1B2) = v(A2|B1A2) = v(A1|A1C2) = v(A2|C1A2). (7.1)

Hence, for these sequences one can write down v(A) as being independent of the sequence. If thecondition is not fulfilled, the model is called contextual. If A is an element of several larger sets ofcompatible observables, the conditions in equation (7.1) can be extended in a straightforward way.

79


Note that the most important assumptions in (7.1) are that v(A1|A1B2) = v(A2|B1A2) andv(A1|A1C2) = v(A2|C1A2). If these are fulfilled, the remaining equality follows from the physicallyvery plausible assumption, that the result of A1 should not depend on which other observable ismeasured after A1 (this observable may be compatible with A or not). It is important to notethat the condition (7.1) is an assumption about the model and — contrary to the definition ofcompatibility — not experimentally testable. This is due to the fact that for a given instance of astate (corresponding to some unknown λ) the experimenter has to decide whether to measure Aor B first.

In the definition, condition (7.1) is assumed for any λ. One may wonder whether this requirementis too strong, since in any experiment only certain λexp can be tested. Assuming, however, thatthere are certain λ which can never occur as a λexp does not make much sense, as in such a HVtheory these λ cannot affect any experimentally observed measurement result, as any pexp(λ) hasto assign a vanishing probability density to them. Hence, one can disregard this case. Finally,let us add here that if larger sets of compatible observables are considered, definitions (1) and(2) have to be extended to longer sequences. For instance, if the observables in each of the setsA,B, C and A, a, α are all compatible, then, the condition that the assignment of a value toA is independent of the context results in conditions similar to equation (7.1), but with sequencesof length three.

v(A1|A1B2C3) = v(A2|B1A2C3) = v(A1|A1a2α3) = v(A2|α1A2a3) = . . . (7.2)

7.1.5. The Kochen Specker inequality for noncontextual models

The KS theorem states that noncontextual HV models cannot reproduce all predictions of QM.Here one (out of many) possible inequalities is discussed involving compatible measurements, whichholds for the QM case but is violated by noncontextual HV theories.

Let’s consider the following array (Mermin Peres square [161, 162]) of observables of a four-levelquantum system (for instance two spin- 1

2 -particles)

A = σ(1)z ⊗ I(2) B = I(1) ⊗ σ

(2)z C = σ

(1)z ⊗ σ

(2)z

a = I(1) ⊗ σ(2)x b = σ

(1)x ⊗ I(2) c = σ

(1)x ⊗ σ

(2)x

α = σ(1)z ⊗ σ

(2)x β = σ

(1)x ⊗ σ

(2)z γ = σ

(1)y ⊗ σ

(2)y

(7.3)

and the measurement products of each row Rk and column Ck given by

R1 = v(A) · v(B) · v(C) C1 = v(A) · v(a) · v(α)R2 = v(a) · v(b) · v(c) C2 = v(B) · v(b) · v(β)R3 = v(α) · v(β) · v(γ) C3 = v(C) · v(c) · v(γ).

(7.4)

Here, σ(k)i denotes the Pauli matrix acting on the k-th particle, and all the observables have the

outcomes ±1. Moreover, in each of the rows or columns of (7.3), the observables are compatible.In any row or column, their measurement product Rk or Ck equals 1, except for the third columnwhere it equals −1. Therefore, quantum mechanics yields for the product

∏k=1,2,3 RkCk a value

of −1, in contrast to non-contextual models.

80

7.2 Experimental realization of QND measurements

Figure 7.1.: Experimental measurement scheme. a For the measurement of the jth row(column) of the Mermin-Peres square (7.3), a quantum state is prepared on which threeconsecutive QND measurements Mk, k = 1, 2, 3, are performed measuring the observablesAjk (Akj). Each measurement consists of a composite unitary operation Uk that mapsthe observable of interest onto one of the single-qubit observables σ

(1)z or σ

(2)z which are

measured by fluorescence detection. The unitary operations U are synthesized from single-qubit and maximally entangling gates. In the lower line, σi, σi symbolize, respectively, theHamiltonian acting on the qubit and on the D5/2 Zeeman subspace spanned by |↓〉, |a〉 =S1/2,mj = 1/2, D5/2,mj = 5/2 which is used for hiding one ion’s quantum state fromfluorescence light during detection[49].

To test this property, it has to be expressed as an inequality since no experiment yields idealquantum measurements. Recently, it has been shown that the inequality

〈XKS〉 = 〈R1〉+ 〈R2〉+ 〈R3〉+ 〈C1〉+ 〈C2〉 − 〈C3〉 ≤ 4 (7.5)

holds for all non-contextual theories [159], where 〈· · · 〉 denotes the ensemble average. Quantummechanics predicts for any state that 〈XKS〉 = 6, thereby violating inequality (7.5). For an experi-mental test, an ensemble of quantum states Ψ needs to be prepared and each realization subjectedto the measurement of one of the possible sets of compatible observables. Here, it is of utmostimportance that all measurements of Aij are context-independent [159], i.e., Aij must be detectedwith a quantum non-demolition (QND) measurement that provides no information whatsoeverabout any other compatible observable.

7.2. Experimental realization of QND measurements

As pointed out in the preceding theory section QND measurements are necessary to test the KochenSpecker theorem. The two qubit observables listed in table 7.3 can be measured in a non destructiveway by applying a unitary transformation U to the state Ψ prior to measuring σz on one ion, andits inverse operation U† after the measurement (see chapter 2.1.3). The gate decompositions ofthe unitary operations necessary to perform all two qubit observables for the Mermin Peres squareare given in table 7.1. These pulse sequences will be used in the chapter 7.3 to determine allobservables for a test of contextual hidden variable theories. The decompositions of Uij into theelementary gate operations available in our setup were found by a gradient-ascent based numericalsearch routine [135]. The unitary transformations are designed such that the information aboutthe observable is mapped onto a single ion (in this case ion 1). The state detection of ion 1 onlyensures that a single bit of information is acquired. The final inverse operation guarantees that

81


U [σx ⊗ σx] = UMSy (−π

2 )U1z (π

2 ) U [σx ⊗ σz] = UMSx (−π

2 )Ux(π2 )

U [σy ⊗ σy] = UMSx (−π

2 )U1z (π

2 ) U [σz ⊗ σx] = UMSy (−π

2 )Ux(−π2 )

U [σz ⊗ σz] = UMSy (−π

2 )U1z (π

2 )Uy(π2 )

Table 7.1.: Gate decomposition of the mapping operations U [σi ⊗ σj ] employed for mea-suring the five two-qubit spin correlations in the Mermin Peres square.

the two ions are in an eigenstate of the measurement.

It is important to note that some of the observables listed in table 7.3 require unitary transfor-mations that are two qubit entangling gates (see table 7.1). As it turns out the most complicatedseries of measurements (row 3 and column 3 in the Mermin Peres square) requires a total of 6two-qubit gate operations. This, by it self, is already quite demanding but gets even more difficultby the fact that the state of one ion has to be detected in between the unitary transformations.The problem that arises with state detection is that the ions do not remain in the ground state butget heated to the Doppler limit. Fortunately the Mølmer-Sørensen gate operation is insensitive tothe motional state of the ions (see chapter 6.1). This enabled us to omit complicated re-coolingschemes, as demonstrated in [55], in the experimental realization of the QND measurements.

The measurement scheme that has to be employed to measure one row or column of the MerminPeres square can be seen in figure 7.1. In the following this scheme will be explained in more detailby an example where (|↓〉 − |↑〉)1 ⊗ (|↓〉 − |↑〉)2 is created as an input state and the last row in theMermin Peres square is measured.

For this example and the experiment described in the following section, a pair of 40Ca+ ions istrapped in a linear Paul trap with axial and radial vibrational frequencies of ωax = (2π) 1.465 MHzand ωr ≈ (2π) 3.4 MHz in a magnetic field of B = 4 Gauss. The ions are Doppler-cooled byexciting the S1/2 ↔ P1/2 and P1/2 ↔ D3/2 dipole transitions. Optical pumping initializes an ionwith a fidelity of 99.5% to the qubit state | ↓〉 ≡ |S1/2,mj = 1/2〉, the second qubit state being|↑〉 ≡ |D5/2,mj = 3/2〉.

The unitary operation to create the input state (|↓〉 − |↑〉)1 ⊗ (|↓〉 − |↑〉)2 is a π/2-pulse onboth ions UΨ = Uy(π

2 ). After the state preparation the first QND measurement, determining theparity σz ⊗ σz of the input state, has to be carried out. This can be done by applying a unitarytransformation which is equivalent to a CNOT operation (see chapter 2.1.3) followed by a PMTdetection of ion 1. This unitary operation mapps the parity of the state on ion 1 and is given interms of gate operations available in the experiment: U1 = UMS

y (−π2 )U1

z (π2 )Uy(π

2 ) which can bewritten in matrix representation as

U1 =

1 0 0 10 1 −1 0i 0 0 −i

0 i i 0

. (7.6)

This operation maps the input state to |↓↓〉 − i |↑↑〉. Detecting the state of ion 1 will resultin equally finding |↓〉 and |↑〉. |↓〉 will be identified with the +1 eigenstate and |↑〉 with the -1

82

7.3 Testing the Kochen Specker inequality

eigenstate. By averaging over many realization the measurement shows that the parity of theinput state is zero.

State detection of ion 1 is achieved by fluorescence detection. Ion 2 is hidden from the fluorescencelaser by transferring the population in the state |↓〉 = S1/2,mj = 1/2 to the D5/2,mj = 5/2 state.The hiding is done by using a pulse sequence similar to the one shown in figure 5.4(b). The onlydifference is that the π

4 pulses are replaced by π2 pulses such that the pulse sequence reads as

U(2)x (π) = Ux(π

2 )U (1)z (π)Ux(π

2 ). The same series of pulses is used to un-hide ion 2 after the statedetection. Additional information on the hiding procedure can be found in appendix A.2

The inverse unitary operation to create an eigenstate of the measurement is achieved by ap-plying the pulse sequence U†

1 = U−y(π2 )U1

z (π2 )UMS

y (−π2 )U1

z (π) which can be written in matrixrepresentation as

U†1 =

i 0 1 00 i 0 10 −i 0 1i 0 −1 0

. (7.7)

If ion 1 was detected in the state |↓〉 this operation maps the two ions onto the entangled state|↑↑〉+|↓↓〉. Similarly if ion 1 was detected in the state |↑〉 the two ions are mapped onto the state|↑↓〉+|↓↑〉. Subsequently the other two QND measurements are performed in the same way. Withthis example one can clearly see that entanglement is created during the measurement process eventhough the initial state was a product state.

7.3. Testing the Kochen Specker inequality

Equipped with the tools described in chapter 5.4 and 6.1, the singlet state Ψ = |↑↓〉 − |↓↑〉 /√2was created by applying the gate-operations U

(1)z (π)U(π

2 , 3π4 )UMS(π

2 , 0) to the initial state |↓↓〉.Then the three observables of a row or column of the Mermin-Peres square were measured con-secutively (as describe in chapter 7.2). The results obtained for a total of 6,600 copies of Ψ arevisualized in figure 7.2. The three upper panels show the distribution of measurement resultsv(Ai1), v(Ai2), v(Ai3) and their products for the observables appearing in the rows of (7.3), thethree lower panels show the corresponding results for the columns of the square. Subplot f demon-strates that Ψ is a common eigenstate of the observables σ

(1)x ⊗ σ

(2)x , σ

(1)y ⊗ σ

(2)y , σ

(1)z ⊗ σ

(2)z , as

only one of the spheres has a considerable size. Five of the correlations have a value close to+1 whereas 〈C3〉 = −0.91(1). By adding them up and subtracting 〈C3〉, one finds a value of〈XKS〉 = 5.46(4) > 4, thus violating equation (7.5).

To test the prediction of a state-independent violation, the experiment was repeated for nineother quantum states of different purity and entanglement. Figure 7.3 shows that indeed a state-independent violation of the Kochen-Specker inequality occurs, 〈XKS〉 ranging from 5.23(5) to5.46(4). It was also checked that a violation of (7.5) occurs irrespective of the temporal order ofthe measurement triples.

Figure 7.4 shows the results for all possible permutations of the rows and columns of (7.3) basedon 39,600 realizations of the singlet state. When combining the correlation results for the 36possible permutations of operator orderings in equation (7.3), all values for the Kochen-Specker

83


σz⊗ Ι

σz⊗

σz

-0.92

0.000.00

Ι ⊗ σ

z Ι ⊗ σx

σx⊗

σx

-0.94

0.02-0.01

σx ⊗

Ι σz⊗ σx

σy⊗

σy

-0.88

-0.01-0.03

σx ⊗

σz

σz⊗ Ι

σz⊗

σx

-0.01

0.010.06

Ι ⊗ σ

x Ι ⊗ σz

σx⊗

σz

-0.10

-0.07-0.03

σx ⊗

Ι σz⊗ σz

σy⊗

σy

-0.89

-0.92-0.95

σx ⊗

σx

⟨R1⟩ = 0.92(1) ⟨R

2⟩ = 0.93(1) ⟨R

3⟩ = 0.90(1)

⟨C1⟩ = 0.90(1) ⟨C

2⟩ = 0.89(1) ⟨C

3⟩ = -0.91(1)

a b c

d e f

Figure 7.2.: Measurement correlations for the singlet state. a This subplot visualizes theconsecutive measurement of the three observables A11 = σ

(1)z , A12 = σ

(2)z , A13 = σ

(1)z ⊗ σ

(2)z

corresponding to row 1 of the Mermin-Peres square. The measurement is carried out on1,100 preparations of the singlet state. The volume of the spheres on each corner of the cuberepresents the relative frequency of finding the measurement outcome v1, v2, v3, vi ∈ ±1.The exact probabilities for finding v1, v2, v3 are given in appendix A.1 . The color of thesphere indicates whether v1v2v3 = +1 (green) or −1 (red). The measured expectation valuesof the observables A1j are indicated by the intersections of the shaded planes with the axes ofthe coordinate system. The average of the measurement product 〈R1〉 is given at the top. b-fSimilarly, the other five subplots represent measurements of the remaining rows or columnsof the Mermin-Peres square. Subplot f demonstrates that the singlet state is a commoneigenstate of the observables σ

(1)x ⊗ σ

(2)x , σ

(1)y ⊗ σ

(2)y , σ

(1)z ⊗ σ

(2)z , as only one of the spheres

has a considerable volume. Taking into account all the results, we find 〈XKS〉 = 5.46(4) inthis measurement. Error bars are 1σ.

84

7.3 Testing the Kochen Specker inequality

Figure 7.3.: State-independence of the Kochen-Specker inequality. The Kochen-Speckerinequality was tested for ten different quantum states, including maximally entangled (Ψ1-Ψ3), partially entangled (Ψ4) and separable (Ψ6-Ψ9) almost pure states as well as an entan-gled mixed state (ρ5) and an almost completely mixed state (ρ10). All states are analysedby quantum state tomography which yields for the experimentally produced states Ψ1-Ψ4,Ψ6-Ψ9 an average fidelity of 97(2)%. For all states, one obtains a violation of inequality (7.5)which demonstrates its state-independent character, the dashed line indicating the averagevalue of 〈XKS〉. Error bars are 1σ (6,600 state realizations per data point).

85


0.75

0.85

0.95

1

row 1 row 2 row 3 column 1 column 3column 2

0.9

0.8

Co

rre

latio

ns

Set of observables

Figure 7.4.: Permutation within rows and columns of the Mermin-Peres square. As thethree observables of a set are commuting, the temporal order of their measurements shouldhave no influence on the measurement results. The figure shows the measured absolute valuesof the products of observables for any of the six possible permutations. The scatter in theexperimental data is caused by experimental imperfections that affect different permutationsdifferently. For the measurements shown here, in total 39,600 copies of the singlet state wereused.

inequality 〈XKS〉 fall within the range of 5.22 to 5.49. Because of experimental imperfections,the experimental violation of the Kochen-Specker inequality falls short of the quantum-mechanicalprediction. The dominating error sources are imperfect unitary operations, in particular the entan-gling gates applied up to six times in a single experimental run. The errors are discussed furtherin section 7.5.1

7.4. Including imperfect measurements to close loopholes

All experimental tests of hidden-variable theories are subject to various possible loopholes. Inthis experiment, the detection loophole does not play a role, as the state of the ions is detectedwith near-perfect efficiency. From the point of view of a hidden variable theory, still objectionscan be made: In the experiment, the observables are not perfectly compatible and since theobservables are measured sequentially, it may be that the hidden variables are disturbed (DHV)during the sequence of measurements, weakening the demand to assign to any observable a fixedvalue independently of the context.

Nevertheless, it is possible to derive inequalities for classical non-contextual models, whereinthe hidden variables are disturbed during the measurement process [160]. The starting point toinclude imperfections is a Clauser-Horne-Shimony-Holt (CHSH) type inequality

〈XCHSH〉 = 〈AB〉+ 〈CD〉+ 〈AC〉 − 〈BD〉 (7.8)

one can derive from equation (7.5) by replacing array (7.3) by

I(1) ⊗ I(2) A = I(1) ⊗ σ(2)y B = σ

(1)y ⊗ σ

(2)y

I(1) ⊗ I(2) C = σ(1)x ⊗ I(2) D = σ

(1)x ⊗ σ

(2)x

I(1) ⊗ I(2) I(1) ⊗ I(2) I(1) ⊗ I(2).

(7.9)

86

7.4 Including imperfect measurements to close loopholes

All nine observables fulfill the condition of compatibility but the inequality, similar to Bell tests,is now violated only by specific states. As most of the observables are now equal to the identitymatrix the relevant measurements are A,B, C,D. A noncontextual HV model has to assign a fixedvalue to each measurement, and any such model predicts

| 〈XCHSH〉 | ≤ 2. (7.10)

If we now consider the QM state

Ψ ∝ |↑↑〉+ iγ|↓↑〉+ γ|↑↓〉+ i|↓↓〉 (7.11)

where γ =√

2− 1, we will find for equation 7.8 a value of

〈XCHSH〉 = 2√

2 (7.12)

violating the HV inequality. The choice of observables is not unique and one can transform allobservables by a global unitary transformation and obtain another set. In fact transformations arepossible that lead to a maximal violation for unentangled states.

To deal with the case of imperfect measurements consider a HV model with a probability distri-bution p(λ) and let p[(A+

1 |A1) and (B+1 |B1)] denote the probability of finding A+ if A is measured

first and B+ if B is measured first, where these conditions are put onto the same instance of theHV. This probability is well defined in all HV models of the considered type but it is impossibleto experimentally measure it directly. The aim is now to connect it to probabilities arising insequential measurements, as this allows one to find contradictions between HV models and QM.

First, note that

p[(A+1 |A1) and (B+

1 |B1)] ≤ p[A+1 , B+

2 |A1B2] + p[(B+1 |B1) and (B−

2 |A1B2)]. (7.13)

This inequality is valid because if λ is such that it contributes to p[(A+1 |A1) and (B+

1 |B1)], then ei-ther the value of B stays the same when measuring A1B2 (hence λ contributes to p[A+

1 , B+2 |A1B2])

or the value of B is flipped and λ contributes to p[(B+1 |B1) and (B−

2 |A1B2)]. The first termp[A+

1 , B+2 |A1B2] is directly measurable as a sequence, but the second term is not directly ex-

perimentally accessible.

Let us rewrite

〈AB〉 = 1− 2p[(A+1 |A1) and (B−

1 |B1)]− 2p[(A−1 |A1) and (B+1 |B1)], (7.14)

as the mean value obtained from the probabilities p[(A±1 |A1) and (B±1 |B1)]. Then, using equa-

tion (7.13), it follows that

〈A1B2〉 − 2pflip[AB] ≤ 〈AB〉 ≤ 〈A1B2〉+ 2pflip[AB], (7.15)

where we used pflip[AB] = p[(B+1 |B1) and (B−

2 |A1B2)]+p[(B−1 |B1) and (B+

2 |A1B2)] which can beinterpreted as a probability that A flips a predetermined value of B.

87


Furthermore, using (7.8), we obtain within a HV model

| 〈XCHSH〉 | ≤ 2(1 + pflip[AB] + pflip[CD] + pflip[AC] + pflip[BD]). (7.16)

The terms pflip[AB], etc. in inequality (7.16) are not experimentally accessible. Now we will discusshow they can be experimentally estimated when some assumptions on the HV model are made.

In order to obtain an experimentally testable version of inequality (7.16), we will assume that

p [(B+1 |B1) and (B−

2 |A1B2)] ≤ p [(B+1 |B1) and (B+

1 , B−3 |B1A2B3)] = p [B+

1 , B−3 |B1A2B3]. (7.17)

We must stress at this point that we do not assume that the set of HV values giving [(B+1 |B1)

and (B−2 |A1B2)] is contained in the set giving (B+

1 , B−3 |B1A2B3), the assumption only relates

the sizes of these two sets. The motive is the following experimental procedure: Let us assume thatone has a physical state, for which one surely finds B+

1 , if B1 is measured first, but one finds B−2

if the sequence A1B2 is measured. Physically, one would explain this behavior with a disturbanceof the system due to the experimental procedures made when measuring A1. The left hand sideof (7.17) can be viewed as the amount of this disturbance. The right hand side quantifies thedisturbance of B when the sequence B1A2B3 is measured. In real experiments, it can be expectedthat this disturbance is larger than when measuring A1B2, because of the additional experimentalprocedures involved. Note that in real experiments, a measurement of B will also disturb the valueof B itself, as can be seen from the fact that sometimes the values of B1 and B2 will not coincide,if the sequence B1B2 is measured (see chapter 7.5.1).

Assumption (7.17) gives a measurable upper bound on pflip[AB]. One directly has

|〈XDHV 〉| = |〈XCHSH〉| − 2(perr[B1A2B3] + perr[D1C2D3]+

+ perr[C1A2C3] + perr[D1B2D3]) ≤ 2, (7.18)

where we used perr[B1A2B3] = p[B+1 , B−

3 |B1A2B3] + p[B−1 , B+

3 |B1A2B3], denoting the total dis-turbance probability of B when measuring B1A2B3.

The point with this inequality is that if the observable pairs (A,B), (C, D), (A,D), and (B,D)fulfill approximately the condition (i) of definition 1 in the definition of compatibility, the termsperr will become small, and a violation of inequality (7.18) can be observed. To test this inequalitywe prepared the state Ψ and performed the measurements listed in (7.9). We find for the value〈XDHV 〉 = 2.23(5) > 2. This proves that even disturbances of the hidden variables for not perfectlycompatible measurements cannot explain the given experimental data. In principle, our analysisof measurement disturbances and dynamical hidden variable models can be extended to the fullMermin-Peres square. The error terms considering now four measurements in a row look like

pflip[(AB)C] ≤ perr[(AB)C] = p[C+1 C−4 |C1A2B3C4 + p[C−1 C+

4 |C1A2B3C4]. (7.19)

Then, if one writes down the generalized form of inequality (7.5) there are more correction termsthan in inequality (7.18), moreover they occur now with weight 4. On average, these perr termshave to be smaller than 2/48 ≈ 0.0417 in order to allow a violation. As can be seen in the next

88

7.5 Experimental results on imperfect measurements

Table 7.2.: Measurement correlations 〈AiAj |A1 . . . A5〉 between repeated measurements ofA = σz ⊗ I for a maximally mixed state. Observing a correlation of 〈AiAj |A1 . . . A5〉 = αij

means that the probability for the measurement results of Ai and Aj to coincide equals(αij + 1)/2.

αij 2 3 4 51 0.97(1) 0.97(1) 0.96(1) 0.95(1)2 0.97(1) 0.97(1) 0.96(1)3 0.98(1) 0.98(1)4 0.98(1)

Table 7.3.: Measurement correlations 〈AiAj |A1 . . . A5〉 between repeated measurements ofA = σx⊗σx for a maximally mixed state. Observing a correlation of 〈AiAj |A1 . . . A5〉 = αij

means that the probability for the measurement results of Ai and Aj to coincide equals(αij + 1)/2.

αij 2 3 4 51 0.94(1) 0.88(1) 0.82(2) 0.80(2)2 0.93(1) 0.87(2) 0.84(2)3 0.90(1) 0.87(2)4 0.93(1)

section the experimental techniques have to be improved to get in this regime and find a violation.

7.5. Experimental results on imperfect measurements

The fact that the results falls short of the quantum mechanical prediction of 〈XKS〉 = 6 is, aspointed out, due to imperfections in the measurement procedure. These imperfections could beincorrect unitary transformations but also errors occurring during the fluorescence measurement.

An instructive test consists in repeatedly measuring the same observable on a single quantumsystem and analyzing the measurement correlations. Table 7.2 shows the results of five consecutivemeasurements of A = σz ⊗ I on a maximally mixed state based on 1100 experimental repetitions.

As expected, the correlations αi,j = 〈AiAj |A1 . . . A5〉 for j=i+1 are independent of the mea-surement number i within the error bars. However, the correlations αi,j for j=i+k becomesmaller and smaller the bigger k gets. Table 7.3 shows another set of measurements correla-tions 〈AiAj |A1 . . . A5〉 where A = σx ⊗ σx. Here, the correlations are smaller, since entanglinginteractions are needed for mapping A onto σz ⊗ I which is experimentally the most demandingstep.

It is also interesting to compare the correlations 〈A1A3|A1A2A3〉 with the correlations 〈A1A3|A1B2A3〉for an observable B that is compatible with A. For A = σx ⊗ σx and B = σz ⊗ σz, we find〈A1A3|A1A2A3〉 = 0.88(1) and 〈A1A3|A1B2A3〉 = 0.83(2) when measuring a maximally mixedstate, i. e. it seems that the intermediate measurement of B perturbs the correlations slightlymore than an intermediate measurement of A. Similar results are found for a singlet state,where 〈A1A3|A1A2A3〉 = 0.92(1), 〈B1B3|B1B2B3〉 = 0.91(1), but 〈A1A3|A1B2A3〉 = 0.90(1),and 〈B1B3|B1A2B3〉 = 0.89(1). Because of 〈B1B3|B1A2B3〉 = 1− 2perr(B1A2B3), correlations ofthe type 〈B1B3|B1A2B3〉 are required for checking the inequality (7.18) that takes into accountdisturbed HVs.

89


7.5.1. Experimental limitations

There are a number of error sources contributing to imperfect state correlations, the most importantbeing:

(i) Wrong state assignment based on fluorescence data. As described in chapter 4.5 we use a250 µs detection time to determine the state of the ion. The photon count distributions slightlyoverlap. Therefore, if the threshold for discriminating between the dark and the bright state is setproperly the probability for wrongly assigning the quantum state is 0.3%. Making the detectionperiod longer would reduce this error but increase errors related to decoherence of the other ion’squantum state that is not measured.

(ii) Imperfect optical pumping. During fluorescence detection, the ion leaves the computationalsubspace |0〉, |1〉 if it was in state |1〉 and can also populate the state |S1/2, mS = −1/2〉. Toprevent this leakage, the ion is briefly pumped on the S1/2 ↔ P1/2 transition with σ+-circularlypolarized light to pump the population back to |1〉. Due to imperfectly set polarization andmisalignment of the pumping beam with respect to the quantization axis, this pumping step failswith a probability of about 0.5%.

(iii) Interactions with the environment. Due to the non-zero differential Zeeman shift of theion’s states used for storing quantum information, superposition states dephase in the presence ofslowly fluctuating magnetic fields. In particular, while measuring one ion by fluorescence detection,quantum information stored in the other ion dephases. We partially compensate for this effect byspin-echo-like techniques [23] that are based on a transient storage of superposition states in a pairof states having an opposite differential Zeeman shift as compared to the states |0〉 and |1〉1. Asecond interaction to be taken into account is spontaneous decay of the metastable state |0〉 whichhowever only contributes an error of smaller than 0.1%.

(iv) Imperfect unitary operations. The mapping operations are not error-free. This concernsin particular the entangling gate operations needed for mapping the eigenstate subspace of a spincorrelation σi⊗σj onto the corresponding subspaces of σz⊗I. For this purpose, a Mølmer-Sørensengate operation UMS(π/2, φ) is used. This gate operation has the crucial property of requiring theions only to be cooled into the Lamb-Dicke regime. In the experiments, the center-of-mass modeused for mediating the gate interaction is in a thermal state with an average of 18 vibrationalquanta. In this regime, the gate operation is capable of mapping |11〉 onto a state |00〉 + eiφ|11〉with a fidelity of about 98% (see chapter 6.1). Taking this fidelity as being indicative of the gatefidelity, one might expect errors of about 4% in each measurement of spin correlations σi ⊗ σj asthe gate is carried out twice, once before and once after the fluorescence measurement.

These error sources prevented us from testing a generalization of inequality (7.5). A measurementof the correlations 〈B1B3|B1A2B3〉 and 〈C1C4|C1A2B3C4〉 resulted in error terms perr that wereabout 0.06 for sequences involving three measurements and about 0.1 for sequences with fourmeasurements, i. e. more than twice as big as required for observing a violation of (7.5).

1The state of the ion not to be measured is stored in a superposition of D5/2, m = 3/2 and D5/2, m = 5/2 duringthe detection. After remapping the state to D5/2, m = 3/2 and S1/2, m = 1/2 a waiting time is implementedduring which the states partially rephase.

90

8. Quantum simulation of the Dirac

equation

Quantum simulation aims at simulating a quantum system using a controllable laboratory systemwith the same underlying mathematical model. In this way, it is possible to simulate quantumsystems that can neither be efficiently simulated on a classical computer [2] nor easily accessedexperimentally, while allowing parameter tunability over a wide range.

Here, a proof-of-principle quantum simulation of the one-dimensional Dirac equation using asingle trapped ion [117] is demonstrated. The ion is set to behave as a free relativistic quantumparticle. The particle position was measured as a function of time and Zitterbewegung was studiedfor different initial superpositions of positive and negative energy spinor states, as well as thecross-over from relativistic to nonrelativistic dynamics. The high level of control of trapped-ionexperimental parameters makes it possible to simulate elegantly textbook examples of relativisticquantum physics.

8.1. Introduction

The difficulties in observing real quantum relativistic effects have sparked significant interest inperforming quantum simulations of their dynamics. Examples include black holes in Bose-Einsteincondensates [163], and Zitterbewegung for massive fermions in solid state physics [164] none ofwhich have been experimentally realized so far. Graphene is studied widely in connection to theDirac equation [165–167].

The Dirac equation is a cornerstone in the history of physics [168], merging successfully quan-tum mechanics with special relativity, providing a natural description of the electron spin andpredicting the existence of anti-matter [169]. Furthermore, it is able to reproduce accurately thespectrum of the hydrogen atom and its realm, relativistic quantum mechanics, is considered asthe natural transition to quantum field theory. However, the Dirac equation also predicts somepeculiar effects such as Klein’s paradox [170] and Zitterbewegung, an unexpected quivering motionof a free relativistic quantum particle first examined by Schrodinger [171]. These and other predic-tions would be difficult to observe in real particles, while constituting key fundamental examplesto understand relativistic quantum effects. Recent years have seen an increased interest in simu-lations of relativistic quantum effects in different physical setups [117, 163, 164, 172–175], whereparameter tunability allows accessibility to different physical regimes.

Trapped ions are particularly interesting for the purpose of quantum simulation [176–178], asthey allow for exceptional control of experimental parameters, and initialization and read-outcan be achieved with high fidelity. Recently, for example, a proof-of-principle simulation of a

91

Quantum simulation of the Dirac equation

quantum magnet has been performed [63] using trapped ions. The full three-dimensional Diracequation Hamiltonian can be simulated with lasers coupling to the three vibrational eigenmodesand the internal states of a single trapped ion [117]. The setup can be significantly simplified whensimulating the Dirac equation in 1+1 dimensions.

8.2. The Dirac equation

This section closely follows reference [179] to explain the effects observed in the experiment. TheDirac equation for a spin-1/2 particle with rest mass m is given by

i~∂ψ

∂t= (c α · p + βmc2)ψ. (8.1)

Here c is the speed of light, p is the momentum operator, α and β are the Dirac matrices, which areusually given in terms of the Pauli matrices σi [168] and the Identity I2 (as defined in chapter 2.1.2)

αi =

(0 σi

σi 0

), β =

(I2 00 −I2

). (8.2)

Each of the entries in these matrices is again a 2 x 2 matrix with i = x, y, z for the three componentsof the momentum operator. The wave functions ψ are 4-component spinors related on one handto the spin of the particle and on the other hand to positive and negative energy solutions. Theoriginal idea by Dirac was that the negative energy states correspond to anti-particles while positiveenergy states correspond to the particle. The problem with this interpretation is that by radiativeinteraction positive energy states could be transformed into negative energy states and atomswould not be stable. Again Dirac proposed a possible solution [180]. All negative energy statesare filled such that no transition is possible due to the Pauli exclusion principle. If one of thesenegative energy electrons is excited it will leave a hole in this sea of particles with a positive chargerepresenting the anti particle. This interpretation has to be handled with care as a number ofproblems appear. The ground state has an infinitely high negative energy. There is an asymmetrybetween particles and anti-particles and we have a multi particle system. A complete solution ofthese problems is only achieved by using a full quantum electro-dynamics theory.

To describe the realized experiment we restrict ourselves to the 1+1 dimensional case of theDirac equation

i~∂ψ

∂t= HD ψ = (c p σx + mc2 σz)ψ. (8.3)

This restriction does not affect the appearance of the most stunning effects of the Dirac equation,such as Zitterbewegung and the Klein paradox. The two components of the spinor in 1+1 dimensiondo not describe the spin as in one dimension all magnetic fields are pure gauge fields [179] and spindoes not exist. They rather represent the appearance of positive and negative energy states

Ψ(x, t) =

(Ψ1(x, t)Ψ2(x, t)

). (8.4)

92

8.2 The Dirac equation

For a further investigation the Dirac Hamiltonian is rewritten in matrix form

HD =

(mc2 cp

cp −mc2

). (8.5)

This matrix has two eigenvalues E(p) = ±√

p2c2 + m2c4 so there are two types of plane wavesolutions to the Dirac equation u±(p; x, t)

u±(p; x, t) =1√2π

u±(p)eipx/~∓iE(p)/~t (8.6)

with u± the eigenvectors to the respective eigenvalues E(p). As a solution for the Dirac equationwe hence have plane waves with positive and negative energy

HD u±(p; x, t) = ±E(p) u±(p;x, t). (8.7)

To account for a real physical situation, that is to find the particle between x = +∞ and x = −∞with probability 1, we have to demand that the integral

∫ +∞

−∞(|Ψ1(x, t)|2 + |Ψ2(x, t)|2)dx = 1 (8.8)

is equal to one. The same argument holds for the Fourier transformed spinors Ψ1,2(p, t) =F(Ψ1,2(x, t)) reflecting the momentum distribution. The unitary time evolution of the Dirac equa-tion guarantees that the normalization of the Dirac spinor is time-independent. One has to notethough, that this is not necessarily true for the individual components Ψ1,2. What holds true is,that if there is a particle at the beginning their will be a particle in the end. The Dirac equationcannot describe particle creation or annihilation.

Square-integrable wave packets can be obtained by superposition of plane waves

Ψ(x, t) =∫ +∞

−∞(Ψ+(p) u+(p; x, t) + Ψ−(p) u−(p;x, t))dp. (8.9)

The coefficient functions Ψ±(p) can be obtained by a projection of Ψ(p, 0) onto the positive andnegative energy subspace

Ψ±(p) = P±(p)Ψ(p, 0). (8.10)

Ψ(p, 0) is obtained by Fourier transforming the initial wave function Ψ(x, 0). In momentum space,the projection operators are given by

P±(p) =12

(I2 ± cpσx + mc2σz√

c2p2 + m2c4

). (8.11)

Note that in general the projection operators do not project onto the spinor basis states except forthe case where p = 0. Only in this case the projector (8.11) becomes diagonal in the spinor basis.

In the Dirac equation in 1+1 dimensions, as given in equation (8.3), there is only one motionaldegree of freedom and the spinor is encoded in two internal levels, related to positive and negative

93


energy states [117]. The velocity of the free Dirac particle is given by

dx/dt = [x,HD]/i~ = cσx (8.12)

in the Heisenberg picture. For a massless particle [σx,HD] = 0 and hence σx is a constant ofmotion. For a massive particle [σx,HD] 6= 0, and the evolution of the particle is described by [171]

x(t) = x(0) + pc2H−1D t + i ξ(e2iHDt/~ − 1), (8.13)

with ξ = 12~c(σx − pcH−1

D )H−1D . The first two terms represent an evolution linear in time, as

expected for a free particle, while the last oscillating term may induce Zitterbewegung.

Zitterbewegung is understood as an interference effect between the positive and negative energyparts of the spinor and does not appear for spinors that consist entirely out of positive (or negative)energy. Furthermore, it is only present when these parts have significant overlap in position andmomentum space and is therefore not a sustained effect under most circumstances [168]. For afree electron, the Dirac equation predicts the Zitterbewegung to have an amplitude of the orderof the Compton wavelength RZB ∼ 10−12 m and a frequency of ωZB ∼ 1021 Hz and has so farbeen experimentally inaccessible. The real existence of Zitterbewegung, in relativistic quantummechanics and in quantum field theory, has been a recurrent subject of discussion in the lastyears [181, 182].

Note that a particle consisting of positive and negative energy states can have zero averagemomentum but a non-zero average velocity. The average velocity of a wave packet is given by theoperator known from classical relativistic mechanics (see equation (8.13))

vCL(p) = c2p H−1D . (8.14)

This relation depends on the sign of the energy. For a wave packet with negative energy, a negativemomentum corresponds to a positive velocity. Thus for an equal superposition of positive andnegative energy states it is possible that the respective average momenta cancel while the averagevelocities are the same. Exactly this situation is simulated in figure 8.1 of the following chapter.

8.3. Experimental realization

For the simulation, a single 40Ca+ ion is trapped in a linear Paul trap [76] with trapping frequenciesωax = 2π×1.36 MHz axially and ωrad = 2π×3 MHz in the radial directions. Doppler cooling,optical pumping, and resolved sideband cooling on the S1/2 ↔ D5/2 transition in a magnetic fieldof 4 G prepare the ion in the axial motional ground state and in the internal state |S1/2,m = 1/2〉.A narrow linewidth laser at 729 nm couples the states

(01

):= |S1/2,m = 1/2〉 and

(10

):= |D5/2,m =

3/2〉 which we identify as our spinor states. A bichromatic light field resonant with the upper andlower axial motional sidebands of the

(10

) ↔ (01

)transition with appropriately set phases [183]

realizes the first part, proportional to σxp, of the Hamiltonian (see chapter 3.6)

HD = 2η∆Ωσxp + ~Ωσz. (8.15)

94

8.3 Experimental realization

Figure 8.1.: Expectation values 〈x(t)〉 for particles with different masses. The linearcurve (¥) represents a massless particle (Ω = 0) moving with the speed of light given byc = 2ηΩ∆ = 0.052 ∆/µs for all curves. The other curves are for particles with increasingmass moving down from the linear curve. Their Compton wavelengths are given by λC :=2ηΩ∆/Ω = 5.4∆ (H), 2.5∆ (¨), 1.2∆ (•) and 0.6∆ (N), respectively. The solid curvesrepresent numerical simulations. The figure shows Zitterbewegung for the crossover fromthe relativistic 2ηΩ À Ω to the nonrelativistic limit 2ηΩ ¿ Ω. The inset shows fittedZitterbewegung amplitude RZB (¥) and frequency ωZB (•) versus the parameter Ω/ηΩ(which is proportional to the mass). Error bars are 1σ.

The component proportional to σz (equivalent to a light shift) is achieved by detuning the redand blue light field by Ω from resonance1. ∆ =

√~/2mωax is the size of the ground-state wave

function with m the ion’s mass, not to be confused with the mass m of the simulated particle,η = 0.06 is the Lamb-Dicke parameter, p = a†−a

2i~∆ is the momentum operator and a† and a are

the usual raising and lowering operators for the motional state along the axial direction. The firstterm in equation 8.15 describes a state-dependent motional excitation with coupling strength ηΩ,corresponding to a displacement of the ion’s wave packet in the harmonic trap. The parameter Ωis controlled by setting the intensity of the bichromatic light field. The second term is equivalent toan optical Stark shift and arises when the bichromatic light field is detuned from resonance by 2Ω.Equation (8.15) reduces to the 1+1 dimensional Dirac Hamiltonian if we make the identificationsc := 2ηΩ∆ and mc2 := ~Ω. The momentum and position of the Dirac particle are then mappedonto the corresponding quadratures of the trapped ion harmonic oscillator.

In order to study relativistic effects such as Zitterbewegung it is necessary to measure the expecta-tion value of the position operator 〈x(t)〉 of the harmonic oscillator. It has been noted theoreticallythat such expectation values could be measured using very short probe times, without reconstruct-ing the full quantum state [115–117]. As pointed out in chapter 3.6.1, for measuring 〈x〉 of amotional state ρm, we have to (i) prepare the ion’s internal state in an eigenstate of σy, (ii) applya unitary transformation U(τ) that maps information about ρm onto the internal states, and (iii)record the changing excitation by a fluorescence measurement [76] as a function of the probe time

1Both light fields are detuned in the same direction unlike a Mølmer-Sørensen interaction.

95


Figure 8.2.: Evolution of the expectation value 〈x(t)〉 for particles with mass λc = 1.2∆and an electronic state which is an eigenstate of σx (•) (data taken from figure 8.1) oran eigenstate of σy (¥). Zitterbewegung appears again due to interference of positive andnegative energy parts of the state. Even though the simulated particles have the same masstheir behavior is completely different. For the σy eigenstate the initial velocity is zero (seeequation 8.12) whereas the particle in an eigenstate of σx has an initial velocity equal to thespeed of light. The solid curve represents a numerical simulation. Error bars are 1σ.

τ (see chapter 3.6.1). In this protocol, the unitary operator U(τ) = exp(−iηΩpσxxτ/∆), withx = (a† + a)∆ and probe Rabi frequency Ωp, effectively transforms the observable σz into sin kx,with k = 2ηΩpτ/∆, so that 〈x〉 can be determined by monitoring the rate of change of 〈sin kx〉for short probe times (see chapter 3.6.1). Since the Dirac Hamiltonian generally entangles the mo-tional and internal state of the ion, we first incoherently recombine the internal state population in(01

)before proceeding to (i).2 Then, we apply the Hamiltonian generating U with the probe Rabi

frequency set to Ωp = 2π× 13 kHz and for interaction times τ of up to 14 µs, with 1-2 µs steps.The change of excitation was obtained by a linear fit each based on 104 to 3 · 104 measurements.

We simulate the Dirac equation by applying HD for varying amounts of time and for differentparticle masses. In the experiment we set Ω = 2π× 68 kHz, corresponding to a simulated speed oflight c = 0.052 ∆/µs. The measured expectation values 〈x(t)〉 are shown in figure 8.1 for a particleinitially prepared in the spinor state ψ(x; t = 0) = (

√2π2∆)−

12 e−

x2

4∆2(11

)by sideband cooling and

application of a π/2-pulse. Zitterbewegung appears for particles with non-zero mass, obtained byvarying Ω in the range of 0 < Ω ≤ 2π× 13 kHz by changing the detuning of the bichromatic lasers.

We investigate the particle dynamics in the cross-over from relativistic to nonrelativistic dynam-ics. The data in figure (8.1) match well with numerical simulations based on the Hamiltonian (8.15)

2This is done by first shelving the population initially in`01

´to |D5/2, m = 5/2〉 using a Rapid Adiabatic Passage

transfer (RAP). A second RAP transfers the population in`10

´to`01

´. A 100 µs laser pulse at 854 nm transfers

the population in |D5/2, m = 5/2〉 to |P3/2, m = 3/2〉 from which it spontaneously decays to`01

´. The transfer

efficiency is > 99 %, limited by the small branching ratio to the D3/2 state. In the transfer steps a probabilityexists that the motional state of the ion is changed. This probability is however very small due to the smallLamb-Dicke parameter but could be eliminated completely by a separate measurement of the motional states ofthe spinor states

`01

´and

`10

´, at the expense of a longer data acquisition time.

96


Figure 8.3.: (a) Zitterbewegung for a state with non-zero average momentum. Initially,Zitterbewegung appears due to interference of positive and negative energy parts of thestate. As these parts separate, the oscillatory motion fades away. The solid curve representsa numerical simulation. (b) Numerically calculated probability distributions |ψ(x)|2 at thetimes t = 0, 75 and 150 µs (as indicated by the arrows in (a)). The probability distributioncorresponding to the state

(01

)is inverted for clarity. The vertical solid line represents 〈x〉 as

plotted in (a). The two dashed lines are the expectation values for the positive and negativeenergy parts of the spinor. (c) Time evolution for a negative energy eigenstate with λC =1.2∆. Laser pulses create a negative energy spinor with average momentum 〈p〉 = 2.2~/∆.The corresponding initial momentum distribution |ψ(p)|2 is shown in the inset. The solidlines represent a numerical calculation. The curve in (c) shows no Zitterbewegung. (d)Simulated probability distributions |ψ(x)|2 for three different evolution times are shown (asindicated by the arrows in (c)). Error bars are 1σ.

shown as solid lines. The error bars are obtained from a linear fit assuming quantum projectionnoise which dominates over noise caused by fluctuations of control parameters. In addition, thedata were fitted with a heuristic model function of the form 〈x(t)〉 = at + RZB sin ωZBt to extractthe effective amplitude RZB and frequency ωZB of the Zitterbewegung shown in the inset. As theparticle’s initial momentum is not dispersion-free, the amplitude and frequency are only approxi-mate concepts. From these data it can be seen that the frequency grows linearly with increasingmass ωZB ≈ 2Ω, whereas the amplitude decreases as the mass is increased. Since the mass ofthe particle is increased but the momentum and the simulated speed of light remain constant, thedata in figure (8.1) show the crossover from the far relativistic to nonrelativistic limits. Hence,the data confirm that Zitterbewegung vanishes in both limits, as theoretically expected. In thefar relativistic case this is because ωZB goes to zero, and in the nonrelativistic case because RZB

vanishes.

The tools used for simulating the Dirac equation can also be applied to precisely set the initialstate of the simulated particle. A first simple change is to prepare an electronic eigenstate of σy.

The wave function of the particle is given by ψ(x; t = 0) = (√

2π2∆)−12 e−

x2

4∆2(1i

). The evolution of

97


Figure 8.4.: (a) Blue sideband oscillations after a 100 µs application of Hamiltonian (8.15)with Ω = 0 and Ω = 2π × 68 kHz demonstrating the coherence of the displacement. Theoscillations for the coherent state with 〈n〉 = 6.3 die out quickly and revive after 500 µs.The solid line was obtained by simulating the phonon distribution and coherently adding theindividual state evolutions. (b) Blue sideband oscillations for the same parameters exceptλc = 1.2∆. The solid line was again obtained by simulating the phonon distribution andcoherently adding the individual state evolutions.

the expectation values 〈x〉 shown in figure 8.2 for particles with equal mass λc = 1.2∆ but differentelectronic state is quite different. The particle initially in an eigenstate of σy has zero velocity att=0 (see equation (8.12)) whereas the particle in an eigenstate of σx has an initial velocity equalto the speed of light.

The particle in figure 8.3a was given an average initial momentum 〈p(t = 0)〉 = ~/∆ by adisplacement operation with the Hamiltonian H = ~ηΩσxx/∆. The initial state of this particle isbuilt up out of a positive energy component with positive velocity and a negative energy componentwith negative velocity[179]. The positive energy component moves to the right and both spinorstates can be seen to contribute to this part (see appendix B.1), the negative component movesto the left. As long as these parts overlap, Zitterbewegung is observed which dies out as the partsseparate. For an additional illustration how the vanishing of the Zitterbewegung comes about theevolution of the probability distributions are shown in figure 8.3 b. The solid lines are obtained bynumerical simulations. The probability distribution corresponding to the state

(01

)is inverted for

clarity.

It is also possible to initialize the spinor in a pure negative or positive energy state (see ap-pendix B.1). In figure 8.3 c, the time evolution 〈x(t)〉 of a negative energy spinor with averagemomentum 〈p〉 = 2.2~/∆ is shown. The corresponding simulated probability distributions aredisplayed in figure 8.3 d and it can be seen that there is indeed no Zitterbewegung or splitting ofthe wave function which arises only if there are positive and negative energy contributions to thewave function.

An additional test to compare experimentally obtained states with theory can be performed byan excitation on the blue sideband after recombining the internal state population. An excitationon the blue sideband will lead to oscillations that die out quickly but revive after some timedue to an interference of the coherently populated states |↓, n〉. This interference occurs becauseall phonon states are excited from the S1/2 ground state to the D5/2 excited state at the same

98


time however their Rabi frequency is different. The populations will thus oscillate at differentfrequencies between ground and excited state and the sum of these oscillations is quickly decaying.Figure 8.4 (a) shows oscillations on the blue sideband after a 100 µs application of Hamiltonian 8.15acting on the initial spinor state ψ(x; t = 0) = (

√2π2∆)−

12 e−

x2

4∆2(11

). The parameters in the

Hamiltonian where set to Ω = 0 and Ω = 2π× 68 kHz. The displacement created a coherent statewith 〈n〉 = 6.3 corresponding to α = 2.5. Similar curves have already been observed in DavidWinelands group [184]. Figure 8.4 (b) shows blue sideband oscillations for the same initial stateand parameters, except Ω = 2π×7 kHz. The solid lines in both figures were obtained by simulatingthe phonon distribution and summing over it,

ρDD(t) =∞∑

n=0

pn sin2(Ωn,n+1t), (8.16)

with Ωn,n+1 the Rabi frequency for coupling the motional states n and n+1 and pn the occupationprobability of the motional state |n〉.

99


100

9. Summary and Outlook

The main results of this thesis can be divided into three topics: QIP with 40Ca+ and 43Ca+

ions realizing high fidelity entangling operations, a test of a hidden variable theory where the testprocedure relies on QIP techniques and finally the quantum simulation of the Dirac equation. Allexperiments encode the quantum information in Zeeman levels of the S1/2 ground state and themeta-stable D5/2 state of calcium ions.

An entangling gate operation is necessary to meet the DiVincenzo criteria to realize a quantumcomputer. For this purpose a Mølmer-Sørensen entangling operation was implemented creatingBell states with 40Ca+ ions with a fidelity of 99.3%. Assuming that the Bell state fidelity is a goodmeasure for the gate-operation, the errors are in principle low enough to be in the fault tolerantregime, although the overhead in the number of qubits is enormous. A detailed experimentalinvestigation of the gate mechanism identified possible error sources. The high quality of the gateoperation allowed for concatenations of up to 21 individual gate-operations. Furthermore it wasshown for the first time, that this kind of gate operation does not depend on ground state cooling.Bell states with a fidelity of 98.4(2)% were realized with ions cooled to a thermal state with a meanvibrational quantum number 〈n〉 = 18(2).

Implementing the gate operation with three ions was the next obvious step. GHZ states werecreated with a fidelity of 97.9(2)%. This is so far the highest achieved fidelity for a three ionGHZ state. By using an off-resonant tightly focused beam a selective entanglement of two outof three ions was achieved. The off-resonant beam was used to switch on and off the Mølmer-Sørensen interaction on each qubit individually. This scheme is an important building block forrealizing NMR like techniques in ion traps. The ideas developed in Volckmar Nebendahls [150]thesis promise simple algorithms realizing e.g. error correction with 9 pulses or a Toffoli gate with11 pulses.

The knowledge gained with 40Ca+ was transferred to the more complicated level structure of43Ca+. A clever choice of polarization and geometrical dependencies helped to suppress the cou-pling strength of unwanted transitions. This paved the way to create entangled states using 43Ca+

ions with a fidelity of 96.9(3)% despite the presence of spectator states. To protect the susceptiblequantum state against magnetic field fluctuations it was mapped onto the hyperfine clock stateof 43Ca+. The mapping pulses work with a fidelity of better than 99% such that the fidelity ofthe Bell state encoded in the hyperfine qubits was 96.7%. After a storage time of 20 ms the statewas mapped back to the optical qubit and two additional entangling operations were applied. Thefidelity after this sequence was still 87.8%. This result demonstrates the combination of a highfidelity gate operation on an optical qubit with the long coherence times of a hyperfine qubit in asingle realization.

The improvements that have to be incorporated for future QIP experiments are manifold. One

101

Summary and Outlook

of the main goals is to unify all building blocks: high fidelity gates, long coherence times, highfidelity state initialization and read out, in one scalable system. The focus of current technologydevelopment will lie on micro-fabricated traps that provide an environment for ions suitable todo QIP. Along this development progress has to be made on many aspects. All techniques haveto be modified such that they work reliably on a bigger number of ions at the same time. Allexperimental parameters have to be controlled with higher precision e.g. laser intensity, trapfrequencies and magnetic field. Achieving this control of parameters is quite challenging. It mightbe more fruitful to develop and improve alternative methods like coherent control [185] or anencoding in decoherence free subspaces [52] that are more robust against parameter fluctuations.

As it turned out, ion trap systems are a good test bed for fundamental questions in quantumphysics or rather hidden variable theories. For this purpose measurements of the form σi⊗σi wererealized by mapping the state of two qubits on a single qubit prior to the fluorescence measure-ment. The non-demolition measurement was completed by a second operation mapping the stateprojected on |↑〉 or |↓〉 onto an eigenstate of the measurement. This technique allowed us to doa series of quantum non-demolition measurements on a single quantum state. The investigatedhigh fidelity entangling gate working on thermal ions was a necessary prerequisite for this task.This measurement technique was the key ingredient to do, for the first time, a state-independentexperimental test of the Kochen Specker theorem. A violation of the Kochen Specker inequalitywas found with 〈XKS〉 = 5.46(4) > 4 indicating that hidden variable theories cannot explain theobserved effects. The theory predicts that this violation will be found for all possible quantumstates. This prediction was tested by performing the same measurements on 10 different inputstates, finding a violation for all states. A possible loophole, incompatibility of measurements,was closed by carefully evaluating possible error sources and including them in a refined theory.We found that quantum mechanics violates a CHSH like inequality including the errors such that〈XDHV 〉 = 2.23(5) > 2. For a violation of the original Kochen Specker inequality incorporatingimperfect measurements the errors of the mapping operations have to be improved by more thana factor of two.

Due to the huge success of ion traps with QIP one tends to forget that an ion trap is “the“model system for a quantum harmonic oscillator. In QIP applications it serves as a mere mediatorto couple different ions, not exploiting the full available Hilbert space. In this thesis the harmonicoscillator was utilized to realize a simulation of the Dirac equation. Momentum and position ofa free Dirac particle were mapped onto the respective observables of the trapped ion harmonicoscillator. The exact Dirac Hamiltonian was implemented and effects like Zitterbewegung could bemeasured. A new method was used to determine the average position of the wave packets. Theamount of control of the system allowed for a preparation of specific states where the form of theZitterbewegung is changed, Zitterbewegung dies out or does not occur at all.

As it turned out the observable A(t) = cos(2ηΩpt/∆x)σz +sin(2ηΩpt/∆x)σy used for measuring〈x〉 can be adopted to reconstruct the probability distribution of the wave function [78, 186]. Thispossibility increases the information about the evolution of the system. It nicely visualizes thesplitting of the wave packet and will be a valuable tool for future simulations.

Possible extensions of the Dirac equation simulation are to add more spatial dimensions or toincorporate potentials to simulate e.g. Klein’s paradox. An interesting example is a Dirac particle

102

on a slope. The Hamiltonian in 1+1 dimensions reads as

Hpot = cpσx + mc2σz + caxI = HD + caxI. (9.1)

The additional potential cax could be realized with a second ion in the trap. HD will be imple-mented with the first ion using the very same light fields as already demonstrated. A displacementoperator on the second ion working on different electronic levels but on the same motional modecreates the desired potential. All that is needed in addition to the presented work is a secondbichromatic light field tuned to a different transition. Simulations suggest that a particle runningup this slope could tunnel through the potential by switching to the negative energy eigenstate.This eigenstate sees an inverted slope and is thus accelerated away from the turning point into theclassically forbidden region. This effect corresponds to Klein’s paradox. Another proposal thatcould be simulated using similar interactions is the 2+1 dimensional Dirac oscillator [172].

Several other proof of principle quantum simulation experiments have been demonstrated inthe last years [63, 186, 187]. So simulations of simple quantum systems are already feasible inion traps. Besides the already realized experiments a huge variety of quantum simulations hasbeen proposed [178] for trapped ions. Some examples are the Unruh effect [188, 189], spin mod-els [177], spin boson models [190], cosmological particle creation [191] and the Frenkel-Kontorovamodel [192]. It seems that in the near future more complicated experimental realizations with afew tens of ions are within reach. Such problems would already be hard to simulate with classicalcomputers.

103

Summary and Outlook

104

A. The Kochen Specker measurements

A.1. Detailed results for the Kochen Specker measurements

The measurement correlations shown in figure 7.2 are based on experiments where measurementson a total of 6,600 realizations of the quantum state Ψ = 1√

2(| ↑↓〉 − | ↓↑〉) were performed. In

these experiments, the results were observed (v1, v2, v3), vi ∈ −1, +1, with the frequencies listedin the table.

M1 σ(1)z σ

(2)x σ

(1)z ⊗ σ

(2)x

Observables M2 σ(2)z σ

(1)x σ

(1)x ⊗ σ

(2)z

M3 σ(1)z ⊗ σ

(2)z σ

(1)x ⊗ σ

(2)z σ

(1)y ⊗ σ

(2)y

(−1,−1,−1) 17 3 20(+1,−1,−1) 520 516 495(−1,+1,−1) 511 529 511

Measured (+1,+1,−1) 8 18 8correlations (−1,−1, +1) 9 15 23

(+1,−1, +1) 4 7 16(−1,+1, +1) 13 10 11(+1,+1, +1) 18 2 16

Number of experiments 1100 1100 1100

M1 σ(1)z σ

(2)z σ

(1)z ⊗ σ

(2)z

Observables M2 σ(2)x σ

(1)x σ

(1)x ⊗ σ

(2)x

M3 σ(1)z ⊗ σ

(2)x σ

(1)x ⊗ σ

(2)z σ

(1)y ⊗ σ

(2)y

(−1,−1,−1) 11 13 1010(+1,−1,−1) 281 298 8(−1,+1,−1) 250 276 14

Measured (+1,+1,−1) 11 18 9correlations (−1,−1, +1) 237 264 26

(+1,−1, +1) 16 13 11(−1,+1, +1) 17 14 22(+1,+1, +1) 277 204 0

Number of experiments 1100 1100 1100

105

The Kochen Specker measurements

Figure A.1.: Sequence used in the Kochen Specker measurements to hide one ion anddetect the state of the other ion. The waiting time counteracts stepwise magnetic fieldfluctuations.

A.2. Hiding the second ion in the Kochen Specker

measurements

The QND measurements performed in chapter 7 required the hiding of the S-state of the secondion from the fluorescence laser in the D5/2,mj = 5/2 state. The hiding was done by using a pulsesequence similar to the one shown in figure 5.4(b) using an addressed AC stark pulse. The pulsesequence is given by U

(2)x (π) = Ux(π

2 )U (1)z (π)Ux(π

2 ). The un-hiding after the detection is done inthe same way. The phases of the two carrier pulses φ that un-hide the ion have to be corrected forthe phase difference the state D5/2,mj = 5/2 acquires relative to the S1/2,mj = 1/2 state. Thewhole pulse sequence for a Kochen Specker measurement including the detection, un-hiding andunitary transformations can be seen in figure A.1.

One problem we were facing in these measurements was a slow sometimes stepwise variationof the magnetic field. The slow variations are tracked by our automatic measurement schemedetermining the magnetic field and laser drift (see chapter 5.3). If a step occurs in the magneticfield by ∆B then the carrier pulses are off-resonant for some time until the change is detected (inthe worst case this can be up to a minute). Due to the detuning the unitary operations followingthe un-hiding will have the wrong phase. This problem can be reduced by the waiting time in thepulse sequence.

The phase difference the qubit acquires when stored in the D-states is given by

φDD′ = µB · gD ·∆B ·∆ t (A.1)

(see equation 2.41) where ∆t is the time needed for state detection and optical pumping. Thephase the qubit acquires during the waiting time ∆t′ is given by

φSD = µB · (gS · 12− gD · 3

2)∆B ·∆t′. (A.2)

The total phase the qubit acquires is thus given by φtotal = φSD + φDD′ . It turns out that thisphase can be canceled by choosing

∆t′ =gD

3/2gD − 1/2gS∆t. (A.3)

This trick only works if the laser is more stable than the magnetic field as the length of the wholepulse sequence is increased. The best compromise in our case was achieved by choosing ∆t = ∆t′

which results in a partial compensation of the phase.

106

B. Methods for simulating the Dirac

equation

B.1. Constructing a pure negative energy spinor

The spinor state in figure 8.3 c was engineered backwards by projecting out the negative energypart of a wavepacket with average momentum 〈p〉 = 2.2~/∆ and renormalizing the spinor.

The complete sequence for approximating the negative energy state is conveniently describedin the basis of the eigenstates |±〉y = 1√

2

(1±i

)of σy. After ground state cooling we prepare

the state |+〉y, then we displace this state to an average momentum state 〈p〉 = 2.2~/∆ by thedisplacement Hamiltonian H = ~ηΩσyx/∆. Next, a far detuned laser pulse rotates the internalstate to 0.84|+〉y + i0.53|−〉y. The displacement Hamiltonian H = −~ηΩσyx/∆ shifts these partsin opposite directions to create the required asymmetry between the average momenta of thecomponents. A final π/2-pulse creates the state shown in figure 8.3 c. This state has > 99%overlap with the desired negative energy state.

B.2. Setting the phase of the Displacement pulse

Two phases φ+, φ− have to be set (see Hamiltonian 3.37) to get the right electronic and motionalinteraction in the displacement Hamiltonian (8.3). The sum phase φ+ can be set by the overallphase of the light field (see figure 4.7 AO2 ). The right value in the massless case (Ω = 0) canbe found by a simple experiment. Applying a Uy(π/2) pulse to the ion we create an eigenstateof σx. If we then apply Hamiltonian 8.3 with Ω = 0 the electronic state of the qubit should notchange. This was confirmed by a second Uy(π/2) pulse completing the rotation to the excitedstate. By scanning the overall phase of the bichromatic light field we can find the phase value thatcorresponds to a σx interaction and implements the desired Hamiltonian.

In the case where Ω 6= 0 (simulation of a particle with mass) finding the right phase was a littlebit more complicated. We again perform the same experiment, Uy(π/2) - apply HD - Uy(π/2) andscan the length of the bichromatic pulse. If the phase is set correctly, we find a change of excitationas shown in figure B.1. The parameters of the displacement pulse were set to Ω = 2π×68 kHz andΩ = 2π× 13 kHz. Comparing these data with simulations (also shown in figure B.1) we found thatthe right phase can be determined by varying the phase φ+ and minimizing the excitation aftera 40 µs bichromatic pulse. The difference phase φ− can be set by changing the relative phases ofthe RF-signal generators driving AO3 (see figure 4.7). Setting the absolute value of φ− of the redand blue light field is not necessary as we can define our phase space such that the displacementwe do is along x. To get a displacement along p we have to add an additional phase difference

107

Methods for simulating the Dirac equation

0 50 100 150

0.6

0.7

0.8

0.9

1.0

Exc

itatio

n to

D5/

2

Displacement length (µs)

Figure B.1.: Excitation evolution for a pulse length scan of the bichromatic pulse in theexperiment Uy(π/2) - apply HD - Uy(π/2) (•). The parameters of the bichromatic pulsewere Ω = 2π × 68 kHz and Ω = 2π× 13 kHz. Here the phase of the bichromatic pulse wasset right realizing 8.3. The right phase can be determined by minimizing the excitation aftera 40 µs pulse length. The red line is a simulation based in 8.3.

of 90 to φ− while keeping the sum frequency φ+ the same. This 90 phase shift of φ− can beachieved by introducing a switchable delay line in the RF line supplying the blue light field. Atthe same time we have to change the phase of the light field by 90 to maintain the same valuefor φ+ and thus the same electronic interaction. A more sophisticated way to change the motionalinteraction in a continuous way from x to p can be achieved by opening up the axial confinement.When we change the trap frequency by an amount of ∆ we have to replace the operators a and a†

in equation (3.38) by aei∆t and a†e−i∆t. So phase space will start to rotate with respect to theunchanged laser frequency that serves as a reference oscillator. If we change the axial trappingpotential by ∆, wait for an appropriate amount of time t and switch back to the initial confinementwe have rotated phase space by ∆t.

A fast switching of the axial trap frequency was achieved with the setup used for shifting theions (see chapter 4.1) but supplying both tips from one of the outputs of the circuit shown infigure 4.2. We can change the trap frequency by about 6 kHz which corresponds to a switchingtime from x to p of about 40 µs.

108

C. Physical and optical properties of

Calcium

speed of light c 299 792 458 m/s (exact)permeability of vacuum µ0 4 π · 107 (exact)permittivity of vacuum ε0 1/(µ0c

2) (exact)Planck’s constant h = 2π~ 6.626 068 96(33) ·10−34 J selementary charge e 1.602 176 487(40) ·10−19 CBohr magneton µB 927.400 915(23) ·10−26 J T−1

atomic mass unit u 1.660 538 782(83)·10−27 kgelectron mass me 9.109 382 15(45)·10−31 kg

fine structure constant α 7.297 352 5376(50)·10−3

Boltzmann constant kB 1.380 6504(24)·10−23

Table C.1.: Fundamental physical constants relevant to the experiment (2006 CODATArecommended values)

transition λair(nm) Isotope shift (MHz) reference4S1/2 ↔ 3D5/2 411.042 129 776 393.3(1.0) THz 4134.711 720 (390) [94, 122, 193]4S1/2 ↔ 3D3/2 732.389 4145(43) [194]4S1/2 ↔ 4P1/2 396.847 706(42) [194]4S1/2 ↔ 4P3/2 393.366 713(31) [194]3P1/2 ↔ 3D3/2 866.214 -3464.3(3.0) [100]3P3/2 ↔ 3D3/2 849.802 -3462.4(2.6) [100]3P3/2 ↔ 3D5/2 854.209 -3465.4(3.7) [100]

Table C.2.: 40Ca+ wavelength in air (λair) and the corresponding isotope shifts (IS)between 40Ca+ and 43Ca+. The transition frequency for 4S1/2 ↔ 3D5/2 is given in THz.

quantity value referencelife time 3D3/2 1.20(1) s [96]life time 3D5/2 1.168(7) s [96]life time 4P1/2 7.098(20) ns [97]life time 4P3/2 6.924(19) ns [97]

gj(4S1/2) 2.00225664(9) [195]gj(3D5/2) 1.200 334 0(3) [122, 193]

Table C.3.: Atomic properties of calcium

109

Physical and optical properties of Calcium

m′ -3/2 -1/2 1/2 3/2 5/2coefficient

√1/5

√2/5

√3/5

√4/5 1

Table C.4.: Clebsch Gordan Coefficients for the 40Ca+ quadrupole transition S1/2,m =1/2 → D5/2, m

′. The coupling strength for S1/2,m = −1/2 → D5/2, m′ can be derived by

exchanging m′j → −m′

j .

0.24

0.17

0.11

0.06

0.03

0.17

0.18

0.15

0.11

0.06

0.11

0.15

0.17

0.15

0.11

0.06

0.11

0.15

0.18

0.17

0.03

0.06

0.11

0.17

0.24

0.34

0.30

0.19

0.09

0.28

0.10

0.26

0.26

0.17

0.31

0.08

0.14

0.26

0.24

0.29

0.20

0.00

0.20

0.29

0.24

0.26

0.14

0.08

0.31

0.17

0.26

0.26

0.10

0.28

0.09

0.19

0.30

0.34

0.48

0.41

0.22

0.41

0.12

0.39

0.33

0.22

0.39

0.14

0.27

0.40

0.33

0.27

0.29

0.09

0.42

0.40

0.09

0.34

0.09

0.40

0.42

0.09

0.29

0.27

0.33

0.40

0.27

0.14

0.39

0.22

0.33

0.39

0.12

0.41

0.22

0.41

0.48

0.68

0.48

0.53

0.31

0.57

0.31

0.53

0.00

0.57

0.12

0.44

0.40

0.23

0.52

0.23

0.50

0.22

0.39

0.44

0.34

0.48

0.00

0.48

0.34

0.44

0.39

0.22

0.50

0.23

0.52

0.23

0.40

0.44

0.12

0.57

0.00

0.53

0.31

0.57

0.31

0.53

0.48

0.68

1.00

0.58

0.82

0.30

0.70

0.65

0.13

0.47

0.71

0.50

0.04

0.25

0.58

0.67

0.38

0.10

0.38

0.65

0.59

0.27

0.17

0.49

0.67

0.49

0.17

0.27

0.59

0.65

0.38

0.10

0.38

0.67

0.58

0.25

0.04

0.50

0.71

0.47

0.13

0.65

0.70

0.30

0.82

0.58

1.00

m = 2

m = 1

m = 0

m = -1

m = -2

F´= 6

F´= 5

F´= 4

F´= 3

F´= 23 D2

5/2

m´ 6543210-6 -5 -4 -3 -2 -1

m = 2

m = 1

m = 0

m = -1

m = -2

m = 2

m = 1

m = 0

m = -1

m = -2

m = 2

m = 1

m = 0

m = -1

m = -2

m = 2

m = 1

m = 0

m = -1

m = -2

Figure C.1.: Coupling strength Λ(F, m, F ′, m′) for the 43Ca+ quadrupole transitionsS1/2(F = 4, m) ↔ D5/2(F ′,m′) with ∆m = m′−m neglecting the geometry and polarizationdependence. Figure taken from [94].

110

D. Journal Publications

The work described in this thesis gave rise to a number of journal publications which are:

1. ”High-fidelity entanglement of 43Ca+ hyperfine clock states.”G.Kirchmair, J.Benhelm, F.Zahringer, R.Gerritsma, C.F.Roos & R.BlattPhys. Rev. A, 79, 020304 (2009)

2. ”Deterministic entanglement of ions in thermal states of motion.”G.Kirchmair, J.Benhelm, F.Zahringer, R.Gerritsma, C.F.Roos & R. BlattNew J. Phys., 11, 023002 (2009)

3. ”State-independent experimental test of quantum contextuality”.G. Kirchmair, F. Zahringer, R. Gerritsma, M. Kleinmann, O.Guhne, A.Cabello, R.Blatt &C.F.RoosNature 460, 494 (2009).

4. ”Quantum simulation of the Dirac equation.”R. Gerritsma, G.Kirchmair, F. Zahringer, E.Solano, R.Blatt & C.F.RoosNature 463, 68 (2010)

Further articles that have been published in the framework of this thesis:

5. ”Measurement of the hyperfine structure of the S1/2-D5/2 transition in 43Ca+.”J.Benhelm, G.Kirchmair, U. Rapol, T.Korber, C.F.Roos & R.Blatt,Phys. Rev. A, 75, 032506 (2007)

6. ”Towards fault-tolerant quantum computing with trapped ions.”J.Benhelm, G.Kirchmair, C.F.Roos & R.BlattNat. Phys. 4, 463 (2008)

7. ”Experimental quantum-information processing with 43Ca+ ions.”J.Benhelm, G.Kirchmair, C.F.Roos & R.BlattPhys. Rev. A 77, 062306 (2008)

8. ”Precision measurement of the branching fractions of the 4P3/2 decay of Ca II.”R. Gerritsma, G. Kirchmair, F. Zahringer, J. Benhelm, R. Blatt, C. F. Roos Eur. Phys. J.D 50,13 (2008)

9. ”Absolute Frequency Measurement of the 40Ca+ 4s2S1/2 − 3d2D5/2 Clock Transition.”M. Chwalla, J. Benhelm, K. Kim, G. Kirchmair, T. Monz, M. Riebe, P. Schindler, A. S. Villar,W. Hansel, C. F. Roos, R. Blatt, M. Abgrall, G. Santarelli, G. D. Rovera, Ph. LaurentPhys. Rev. Lett. 102, 023002 (2009).

111

Journal Publications

10. ”Realization of a quantum walk with one and two trapped ions.”F. Zahringer, G.Kirchmair, R. Gerritsma, E.Solano, R.Blatt & C.F.RoosPhys. Rev. Lett. 104 100503 (2010)

11. ”Compatibility and noncontextuality for sequential measurements.”O. Guhne, M. Kleinmann, A. Cabello, J.-A. Larsson, G. Kirchmair, F. Zahringer, R. Ger-ritsma, C.F. RoosPhys. Rev. A 81, 022121 (2010)

112

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Danksagung

Das Gelingen dieser Arbeit ware ohne die Hilfe und Geduld Vieler nicht moglich gewesen. Ihnenallen mochte hiermit herzlich danken.

An erster Stelle mochte ich mich bei Rainer Blatt bedanken, der es mir ermoglicht hat meineDiplomarbeit und jetzt auch meine Doktorarbeit in seiner Arbeitsgruppe anzufertigen. Durchsein stetiges Bemuhen und seinen großen Einsatz hat er ein Klima und wissenschaftliches Umfeldgeschaffen indem es eine Freude ist zu arbeiten.

Ein großer Dank gebuhrt naturlich Christian Roos. Viele der in dieser Arbeit vorgestelltenExperimente waren nur durch seine Ideen und Unterstutzung moglich. Besonders Bedanken mochteich mich auch fur das Korrekturlesen meiner Arbeit.

Meine Arbeitsweise im Labor wurde am meisten durch Jan Benhelm gepragt. In den vier Jahrenunserer Zusammenarbeit konnte ich sehr viel von ihm lernen und meine Doktorarbeit profitiertesehr von seiner sorgfaltigen Aufbauarbeit.

Weiters Bedanken mochte ich mich bei Rene Gerritsma und Florian Zahringer die mit mirunzahlige Tage im Labor verbracht haben. Die interessanten und witzigen Unterhaltungen wahrendder langen Stunden vor dem Laborrechner werden mir immer in guter Erinnerung bleiben.

Vielen Dank auch an die ganze 40Ca+Crew fur die gute Zusammenarbeit in den letzten Jahren.Besonders bedanken mochte ich mich bei Philipp Schindler fur das betreuen unserer Pulse Boxund Michael Chwalla fur die Arbeit an den 729 nm cavities und der ”real laser front”.

Ein großer Dank gebuhrt auch der ganzen Kochgruppe die, jeden Tag, fur ein sehr gutes Essenund eine gute Stimmung wahrend der Mittagspause gesorgt hat.

Nicht zu vergessen sind naturlich meine Freunde. Einen Physiker von seiner Arbeit abzulenkenist nicht immer ganz einfach deshalb ist es ihnen zu verdanken, dass meine sozialen Kompetenzennicht auf der Strecke geblieben sind.

Ohne die Unterstutzung meiner Familie hatte ich diese Doktorarbeit wohl nicht so erfolgreichabschliessen konnen. Deshalb will ich mich an dieser Stelle auch recht herzlich bei ihnen bedanken.

Ein besonderer Dank gebuhrt meiner Freundin Astrid die mich all die Jahre durch mein Studiumund meine Doktorarbeit begleitet hat. Ihr Verstandnis und ihre Unterstutzung haben mir geholfenauch die schwierigsten Situationen zu meistern.

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quantum non-demolition measurements and quantum simulation · 2015-12-01 · all divincenzo...

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